aa r X i v : . [ m a t h . S G ] D ec Lectures on Symplectic Field Theory
Chris Wendl
Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin, Unterden Linden 6, 10099 Berlin, Germany
E-mail address : [email protected] ontents Preface viiAbout the current version ixLecture 1. Introduction 11.1. In the beginning, Gromov wrote a paper 11.2. Hamiltonian Floer homology 41.3. Contact manifolds and the Weinstein conjecture 91.4. Symplectic cobordisms and their completions 161.5. Contact homology and SFT 201.6. Two applications 23Lecture 2. Basics on holomorphic curves 252.1. Linearized Cauchy-Riemann operators 252.2. Some useful Sobolev inequalities 282.3. The fundamental elliptic estimate 302.4. Regularity 322.5. Linear local existence and applications 382.6. Simple curves and multiple covers 41Lecture 3. Asymptotic operators 433.1. The linearization in Morse homology 433.2. Spectral flow 463.3. The Hessian of the contact action functional 573.4. The Conley-Zehnder index 61Lecture 4. Fredholm theory with cylindrical ends 674.1. Cauchy-Riemann operators with punctures 674.2. A global weak regularity result 704.3. Elliptic estimates on cylindrical ends 714.4. The semi-Fredholm property 734.5. Formal adjoints and proof of the Fredholm property 74Lecture 5. The index formula 815.1. Riemann-Roch with punctures 815.2. Some remarks on the formal adjoint 865.3. The index zero case on a torus 905.4. A Weitzenb¨ock formula for Cauchy-Riemann operators 925.5. Large antilinear perturbations and energy concentration 94 iiiv
Chris Wendl C ε to C ∞ T and Giroux torsion 19110.2. Definition of cylindrical contact homology 19410.3. Computing HC ∗ ( T , ξ k ) 208Lecture 11. Coherent orientations 22511.1. Gluing maps and coherence 22511.2. Permutations of punctures and bad orbits 23011.3. Orienting moduli spaces in general 23211.4. The determinant line bundle 23411.5. Determinant bundles of moduli spaces 23711.6. An algorithm for coherent orientations 23811.7. Permutations and bad orbits revisited 240Lecture 12. The generating function of SFT 24312.1. Some important caveats on transversality 24312.2. Auxiliary data, grading and supercommutativity 244 ectures on Symplectic Field Theory v H and commutators 24712.4. Interlude: How to count points in an orbifold 25112.5. Cylindrical contact homology revisited 25612.6. Combinatorics of gluing 25912.7. Some remarks on torsion, coefficients, and conventions 263Lecture 13. Contact invariants 26713.1. The Eliashberg-Givental-Hofer package 26813.2. SFT generating functions for cobordisms 27813.3. Full SFT as a BV ∞ -algebra 288Lecture 14. Transversality and embedding controls in dimension four 295Lecture 15. Intersection theory for punctured holomorphic curves 297Lecture 16. Torsion computations and applications 299Appendix A. Sobolev spaces 301A.1. Approximation, extension and embedding theorems 301A.2. Products, compositions, and rescaling 305A.3. Spaces of sections of vector bundles 311A.4. Some remarks on domains with cylindrical ends 316Appendix B. The Floer C ε space 319Appendix C. Genericity in the space of asymptotic operators 323Bibliography 329 reface This book is a slightly expanded version of the lecture notes I produced for atwo-semester course taught at University College London in 2015–16, for Ph.D. stu-dents with a background in basic symplectic geometry and interest in symplectictopology and/or geometric analysis. I say “slightly expanded,” although the readerwill quickly notice that most individual chapters contain far more material than canreasonably fit into a two-hour lecture. In reality, much of that material was onlysketched or mentioned in passing during lectures, and I ended up using the notesto discuss everything that I would like to have explained if I’d had unlimited time.This includes relatively detailed discussions of several important technical points(e.g. the definition of spectral flow, generic transversality in symplectizations, thepunctured Riemann-Roch formula, finite energy and asymptotics with arbitrary sta-ble Hamiltonian structures) which are either incompletely covered by the existingliterature or, in my opinion, simply more difficult to learn from other sources thanthey should be. For topics that are on the other hand well covered elsewhere, I haveusually not felt obliged to explain every detail, but have tried always to provideadequate references.One of the interesting features of SFT is that its foundations are—at the time ofthis writing—not yet complete. When the original “propaganda paper” [
EGH00 ]appeared in 2000, it was widely believed that the technical details would be filled inwithin a few years, and several papers introducing important applications of SFTto contact topology were written under this assumption. Since then, a certain re-alization has set in that the results in those papers cannot truly be regarded as“theorems” in the sense of mathematics, and it has become less socially acceptableto preface statements of results with caveats of the form, “this theorem is dependenton the foundations of SFT”. At the same time, the need for a robust perturbationscheme to achieve transversality in SFT spawned the development of a whole newapproach to infinite-dimensional differential geometry, the polyfold project [
Hof06 ],which is intended for much more general applications but is not yet finished. Opin-ions vary among symplectic topologists as to how unsatisfied we should all be withthis state of affairs, and what could be done about it—among other things, one couldmake an entire course out of the discussion of such issues, but I have not chosen todo that. My approach is instead to develop the classical analysis of pseudoholo-morphic curves in symplectizations and symplectic cobordisms, to explain how thiswould lead to a theory of algebraic contact invariants if transversality for multiplecovers were not an issue, and then to use the tools and insights gained from this For the purposes of this discussion, the word “classical” may be defined as “not involving thewords polyfold , virtual or Kuranishi ”. viiiii Chris Wendl discussion to prove rigorous mathematical theorems about contact manifolds. Typi-cally, such theorems can be regarded informally as consequences of computations ina (not yet well-defined) theory called SFT, but in a rigorous sense, they are actuallyconsequences of the methods used in those computations. Examples covered in thesenotes include distinguishing tight contact structures on the 3-torus that are homo-topic but not isomorphic (Lecture 10), and the nonexistence of symplectic fillingsor symplectic cobordisms between certain pairs of contact manifolds (Lecture 16).The choice of applications is of course biased somewhat toward my own researchinterests.
Prerequisites.
The stated target audience for the lecture course was “Ph.D. stu-dents in differential geometry or related fields who are not afraid of analysis”. Moreprecisely, the notes assume some knowledge of the following topics: • Differential geometry: manifolds and vector bundles, differential forms andStokes’ theorem, connections, basic familiarity with symplectic manifolds • Functional analysis: linear operators on Banach spaces, basics of Sobolevspaces, Fredholm operators • Differential topology: smooth mapping degree, intersection numbers, Sard’stheorem • Algebraic topology: fundamental group, homology and cohomology of man-ifolds, Poincar´e duality, first Chern class, homological intersection numbersThe following topics are not considered formal prerequisites, but some knowledge ofthem is likely in any case to be helpful to the reader, who may want to have a goodreference for them (as suggested below) within arm’s reach: • Contact manifolds (e.g. Geiges [
Gei08 ]) • Differential calculus on Banach spaces and Banach manifolds (e.g. thesetwo books by Lang: [
Lan93 ] and [
Lan99 ]) • Closed pseudoholomorphic curves (e.g. McDuff-Salamon [
MS04 ] or myother book in preparation [
Wend ]) • Floer homology (e.g. Salamon [
Sal99 ] or Audin-Damian [
AD14 ]) Acknowledgements.
I would like to thank the students who sat through thecourse that gave rise to these notes, and in particular Alexandru Cioba and Agust´ınMoreno for their assistance in editing the first several lectures. My understandingof Taubes’s approach to the Riemann-Roch formula (explained in Lecture 5) and itsgeneralization to the punctured case emerged in part from discussions with ChrisGerig, and I am grateful also to Tim Perutz for helpful hints about Weitzenb¨ockformulas, and Patrick Massot for patient discussions of singular integral operatorsand elliptic regularity. Thanks also to Michael Hutchings and Janko Latschev forhelping me understand the combinatorial factors in Lecture 12, to Jo Nelson forhelpful comments on coefficients and orbifold singularities, and to Sam Lisi andBarney Bramham for advice on the Floer C ε space. bout the current version At the time of posting this on the arXiv, Lectures 14, 15 and 16 each consistof messy handwritten notes that have not yet been typed up, but will eventuallyappear in the published version of the book. The main goal for those lecturesis to carry out some explicit computations of the torsion invariant introduced atthe end of Lecture 13, and to explain the consequences for filling and cobordismobstructions, including for instance the classic result that overtwistedness impliesvanishing contact homology and thus obstructs fillability. In keeping with the spiritof the book, the theorems about torsion in Lecture 16 will need to be understoodwith the usual caveat that they depend on the unfinished foundations of SFT, butpart of the point is also to extract complete and rigorous proofs of the importantconsequences regarding symplectic fillings. Lectures 14 and 15 are more technicalin nature, in the spirit of Lectures 2 through 9 except that they deal with topicsthat are only relevant in low-dimensional settings (and thus significantly increasethe power of the theory in those settings). Aside from dealing with topics thatare valuable in their own right, they specifically precede Lecture 16 because theyintroduce techniques that will be used in the computations in that lecture.As far as the rest of the manuscript is concerned, I have tried to produce some-thing that is relatively well polished, but I admit I have not tried quite as diligentlyfor that as I do with most of my research papers. Trying to produce another oneof these lectures every week while teaching the course was a formidable task, and Ihad more time to be careful with it in some weeks than in others. I have since goneback and reworked some portions, but not all, so I apologize for any sloppiness thatI may have failed so far to expunge. All comments and corrections are welcome, and may be sent to [email protected] . Updates on the publication of thebook will be posted periodically on my website at especially if those corrections are received before the book goes to press ix ECTURE 1
IntroductionContents T Symplectic field theory is a general framework for defining invariants of contactmanifolds and symplectic cobordisms between them via counts of “asymptoticallycylindrical” pseudoholomorphic curves. In this first lecture, we’ll summarize someof the historical background of the subject, and then sketch the basic algebraicformalism of SFT.
Pseudoholomorphic curves first appeared in symplectic geometry in a 1985 paperof Gromov [
Gro85 ]. The development was revolutionary for the field of symplectictopology, but it was not unprecedented: a few years before this, Donaldson haddemonstrated the power of using elliptic PDEs in geometric contexts to define in-variants of smooth 4-manifolds (see [
DK90 ]). The PDE that Gromov used was aslight generalization of one that was already familiar from complex geometry.Recall that if M is a smooth 2 n -dimensional manifold, an almost complexstructure on M is a smooth linear bundle map J : T M → T M such that J = − .This makes the tangent spaces of M into complex vector spaces and thus induces anorientation on M ; the pair ( M, J ) is called an almost complex manifold . In thiscontext, a
Riemann surface is an almost complex manifold of real dimension 2(hence complex dimension 1), and a pseudoholomorphic curve (also called J -holomorphic ) is a smooth map u : Σ → M satisfying the nonlinear Cauchy-Riemann equation (1.1) T u ◦ j = J ◦ T u, Chris Wendl where (Σ , j ) is a Riemann surface and (
M, J ) is an almost complex manifold (ofarbitrary dimension). The almost complex structure J is called integrable if M is admits the structure of a complex manifold such that J is multiplication by i in holomorphic coordinate charts. By a basic theorem of the subject, every almostcomplex structure in real dimension two is integrable, hence one can always findlocal coordinates ( s, t ) on neighorhoods in Σ such that j∂ s = ∂ t , j∂ t = − ∂ s . In these coordinates, (1.1) takes the form ∂ s u + J ( u ) ∂ t u = 0 . The fundamental insight of [
Gro85 ] was that solutions to the equation (1.1)capture information about symplectic structures on M whenever they are related to J in the following way. Definition . Suppose (
M, ω ) is a symplectic manifold. An almost complexstructure J on M is said to be tamed by ω if ω ( X, J X ) > X ∈ T M with X = 0 . Additionally, J is compatible with ω if the pairing g ( X, Y ) := ω ( X, J Y )defines a Riemannian metric on M .We shall denote by J ( M ) the space of all smooth almost complex structures on M , with the C ∞ loc -topology, and if ω is a symplectic form on M , let J τ ( M, ω ) , J ( M, ω ) ⊂ J ( M )denote the subsets consisting of almost complex structures that are tamed by orcompatible with ω respectively. Notice that J τ ( M, ω ) is an open subset of J ( M ),but J ( M, ω ) is not. A proof of the following may be found in [
Wend , § Proposition . On any symplectic manifold ( M, ω ) , the spaces J τ ( M, ω ) and J ( M, ω ) are each nonempty and contractible. (cid:3) Tameness implies that the energy of a J -holomorphic curve u : Σ → M , E ( u ) := Z Σ u ∗ ω, is always nonnegative, and it is strictly positive unless u is constant. Notice moreoverthat if the domain Σ is closed, then E ( u ) depends only on the cohomology class[ ω ] ∈ H ( M ) and the homology class[ u ] := u ∗ [Σ] ∈ H ( M ) , so in particular, any family of J -holomorphic curves in a fixed homology class sat-isfies a uniform energy bound. This basic observation is one of the key facts behindGromov’s compactness theorem, which states that moduli spaces of closed curves ina fixed homology class are compact up to “nodal” degenerations. ectures on Symplectic Field Theory The most famous application of pseudoholomorphic curves presented in [
Gro85 ]is Gromov’s nonsqueezing theorem , which was the first known example of an obstruc-tion for embedding symplectic domains that is subtler than the obvious obstructiondefined by volume. The technology introduced in [
Gro85 ] also led directly to thedevelopment of the
Gromov-Witten invariants (see [
MS04 , RT95 , RT97 ]), whichfollow the same pattern as Donaldson’s earlier smooth 4-manifold invariants; theyuse counts of J -holomorphic curves to define invariants of symplectic manifolds upto symplectic deformation equivalence.Here is another sample application from [ Gro85 ]. We denote by A · B ∈ Z the intersection number between two homology classes A, B ∈ H ( M ) in a closedoriented 4-manifold M . Theorem . Suppose ( M, ω ) is a closed and connected symplectic -manifoldwith the following properties:(i) ( M, ω ) does not contain any symplectic submanifold S ⊂ M that is diffeo-morphic to S and satisfies [ S ] · [ S ] = − .(ii) ( M, ω ) contains two symplectic submanifolds S , S ⊂ M which are bothdiffeomorphic to S , satisfy [ S ] · [ S ] = [ S ] · [ S ] = 0 , and have exactly one intersection point with each other, which is transverseand positive.Then ( M, ω ) is symplectomorphic to ( S × S , σ ⊕ σ ) , where for i = 1 , , the σ i are area forms on S satisfying Z S σ i = h [ ω ] , [ S i ] i . Sketch of the proof.
Since S and S are both symplectic submanifolds,one can choose a compatible almost complex structure J on M for which both ofthem are the images of embedded J -holomorphic curves. One then considers themoduli spaces M ( J ) and M ( J ) of equivalence classes of J -holomorphic sphereshomologous to S and S respectively, where any two such curves are consideredequivalent if one is a reparametrization of the other (in the present setting this justmeans they have the same image). These spaces are both manifestly nonempty,and one can argue via Gromov’s compactness theorem for J -holomorphic curvesthat both are compact. Moreover, an infinte-dimensional version of the implicitfunction theorem implies that both are smooth 2-dimensional manifolds, carryingcanonical orientations, hence both are diffeomorphic to closed surfaces. Finally, oneuses positivity of intersections to show that every curve in M ( J ) intersects everycurve in M ( J ) exactly once, and this intersection is always transverse and positive;moreover, any two curves in the same space M ( J ) or M ( J ) are either identicalor disjoint. It follows that both moduli spaces are diffeomorphic to S , and bothconsist of smooth families of J -holomorphic spheres that foliate M , hence defining Chris Wendl a diffeomorphism M ( J ) × M ( J ) → M that sends ( u , u ) to the unique point in the intersection im u ∩ im u . This identifies M with S × S such that each of the submanifolds S × {∗} and {∗} × S aresymplectic. The latter observation can be used to determine the symplectic formup to deformation, so that by the Moser stability theorem, ω is determined up toisotopy by its cohomology class [ ω ] ∈ H ( S × S ), which depends only on theevaluation of ω on [ S × {∗} ] and [ {∗} × S ] ∈ H ( S × S ). (cid:3) For a detailed exposition of the above proof of Theorem 1.3, see [
Wene , Theo-rem E].
Throughout the following, we write S := R / Z , so maps on S are the same as 1-periodic maps on R . One popular version of the Arnold conjecture on symplectic fixed points can be stated as follows. Suppose(
M, ω ) is a closed symplectic manifold and H : S × M → R is a smooth func-tion. Writing H t := H ( t, · ) : M → R , H determines a 1-periodic time-dependentHamiltonian vector field X t via the relation (1.2) ω ( X t , · ) = − dH t . Conjecture . If all -periodic orbits of X t are nonde-generate, then the number of these orbits is at least the sum of the Betti numbersof M . Here a 1-periodic orbit γ : S → M of X t is called nondegenerate if, denotingthe flow of X t by ϕ t , the linearized time 1 flow dϕ ( γ (0)) : T γ (0) M → T γ (0) M does not have 1 as an eigenvalue. This can be thought of as a Morse condition foran action functional on the loop space whose critical points are periodic orbits; likeMorse critical points, nondegenerate periodic orbits occur in isolation. To simplifyour lives, let’s restrict attention to contractible orbits and also assume that ( M, ω )is symplectically aspherical , which means[ ω ] | π ( M ) = 0 . Then if C ∞ contr ( S , M ) denotes the space of all smoothly contractible smooth loopsin M , the symplectic action functional can be defined by A H : C ∞ contr ( S , M ) → R : γ
7→ − Z D ¯ γ ∗ ω + Z S H t ( γ ( t )) dt, Elsewhere in the literature, you will sometimes see (1.2) without the minus sign on the righthand side. If you want to know why I strongly believe that the minus sign belongs there, see[
Wenc ], but to some extent this is just a personal opinion. ectures on Symplectic Field Theory where ¯ γ : D → M is any smooth map on the closed unit disk D ⊂ C satisfying¯ γ ( e πit ) = γ ( t ) , and the symplectic asphericity condition guarantees that A H ( γ ) does not dependon the choice of ¯ γ . Exercise . Regarding C ∞ contr ( S , M ) as a Fr´echet manifold with tangentspaces T γ C ∞ contr ( S , M ) = Γ( γ ∗ T M ), show that the first variation of the action func-tional A H is d A H ( γ ) η = Z S [ ω ( ˙ γ, η ) + dH t ( η )] dt = Z S ω ( ˙ γ − X t ( γ ) , η ) dt for η ∈ Γ( γ ∗ T M ). In particular, the critical points of A H are precisely the con-tractible 1-periodic orbits of X t .A few years after Gromov’s introduction of pseudoholomorphic curves, Floerproved the most important cases of the Arnold conjecture by developing a novelversion of infinite-dimensional Morse theory for the functional A H . This approachmimicked the homological approach to Morse theory which has since been popular-ized in books such as [ AD14 , Sch93 ], but was apparently only known to experts atthe time. In
Morse homology , one considers a smooth Riemannian manifold (
M, g )with a Morse function f : M → R , and defines a chain complex whose generatorsare the critical points of f , graded according to their Morse index. If we denote thegenerator corresponding to a given critical point x ∈ Crit( f ) by h x i , the boundarymap on this complex is defined by ∂ h x i = X ind( y )=ind( x ) − (cid:0) M ( x, y ) (cid:14) R (cid:1) h y i , where M ( x, y ) denotes the moduli space of negative gradient flow lines u : R → M ,satisfying ∂ s u = −∇ f ( u ( s )), lim s →−∞ u ( s ) = x and lim s → + ∞ u ( s ) = y . This spaceadmits a natural R -action by shifting the variable in the domain, and one can showthat for generic choices of f and the metric g , M ( x, y ) / R is a finite set wheneverind( x ) − ind( y ) = 1. The real magic however is contained in the following statementabout the case ind( x ) − ind( y ) = 2: Proposition . For generic choices of f and g and any two critical points x, y ∈ Crit( f ) with ind( x ) − ind( y ) = 2 , M ( x, y ) / R is homeomorphic to a finitecollection of circles and open intervals whose end points are canonically identifiedwith the finite set ∂ M ( x, y ) := [ ind( z )=ind( x ) − M ( x, z ) × M ( z, y ) . We say that M ( x, y ) has a natural compatification M ( x, y ), which has thetopology of a compact 1-manifold with boundary, and its boundary is the set ofall broken flow lines from x to y , cf. Figure 1.1. This set of broken flow linesis precisely what is counted if one computes the h y i coefficient of ∂ h x i , hence wededuce ∂ = 0 Chris Wendl
Figure 1.1.
One-parameter families of gradient flow lines on aRiemannian manifold degenerate to broken flow lines.as a consequence of the fact that compact 1-manifolds always have zero boundarypoints when counted with appropriate signs. The homology of the resulting chaincomplex can be denoted by HM ∗ ( M ; g, f ) and is called the Morse homology of M . The well-known Morse inequalities can then be deduced from a fundamen-tal theorem stating that HM ∗ ( M ; g, f ) is, for generic f and g , isomorphic to thesingular homology of M .With the above notion of Morse homology understood, Floer’s approach to theArnold conjecture can now be summarized as follows: Step 1:
Under suitable technical assumptions, construct a homology theory HF ∗ ( M, ω ; H, { J t } ) , depending a priori on the choices of a Hamiltonian H : S × M → R withall 1-periodic orbits nondegenerate, and a generic S -parametrized familyof ω -compatible almost complex structures { J t } t ∈ S . The generators of thechain complex are the critical points of the symplectic action functional A H , i.e. 1-periodic orbits of the Hamiltonian flow, and the boundary mapis defined by counting a suitable notion of gradient flow lines connectingpairs of orbits (more on this below). Step 2:
Prove that HF ∗ ( M, ω ) := HF ∗ ( M, ω ; H, { J t } ) is a symplectic invariant ,i.e. it depends on ω , but not on the auxiliary choices H and { J t } . Step 3:
Show that if H and { J t } are chosen to be time-independent and H isalso C -small, then the chain complex for HF ∗ ( M, ω ; H, { J t } ) is isomor-phic (with a suitable grading shift) to the chain complex for Morse ho-mology HM ∗ ( M ; g, H ) with g := ω ( · , J t · ). The isomorphism between HM ∗ ( M ; g, H ) and singular homology thus implies that the Floer com-plex must have at least as many generators (i.e. periodic orbits) as thereare generators of H ∗ ( M ), proving the Arnold conjecture. Counting with signs presumes that we have chosen suitable orientations for the moduli spaces M ( x, y ), and this can always be done. Alternatively, one can avoid this issue by counting modulo 2and thus define a homology theory with Z coefficients. ectures on Symplectic Field Theory The implementation of Floer’s idea required a different type of analysis thanwhat is needed for Morse homology. The moduli space M ( x, y ) in Morse homol-ogy is simple to understand as the (generically transverse) intersection between theunstable manifold of x and the stable manifold of y with respect to the negativegradient flow. Conveniently, both of those are finite-dimensional manifolds, withtheir dimensions determined by the Morse indices of x and y . We will see in Lec-ture 3 that no such thing is true for the symplectic action functional: to the extentthat A H can be thought of as a Morse function on an infinite-dimensional manifold,its Morse index and its Morse “co-index” at every critical point are both infinite,hence the stable and unstable manifolds are not nearly as nice as finite-dimensionalmanifolds, providing no reason to expect that their intersection should be. Thereare additional problems since C ∞ contr ( S , M ) does not have a Banach space topology:in order to view the negative gradient flow of A H as an ODE and make use of theusual local existence/uniqueness theorems (as in [ Lan99 , Chapter IV]), one wouldhave to extend to A H to a smooth function on a suitable Hilbert manifold with aRiemannian metric. There is a very limited range of situations in which one can dothis and obtain a reasonable formula for ∇A H , e.g. [ HZ94 , § M = T n , in which A H can be defined on the Sobolev space H / ( S , R n ) and thenstudied using Fourier series. This approach is very dependent on the fact that thetorus T n is a quotient of R n ; for general symplectic manifolds ( M, ω ), one cannoteven define H / ( S , M ) since functions of class H / on S need not be continuous( H / is a “Sobolev borderline case” in dimension one).One of the novelties in Floer’s approach was to refrain from viewing the gradientflow as an ODE in a Banach space setting, but instead to write down a formalversion of the gradient flow equation and regard it as an elliptic PDE. To this end,let us regard C ∞ contr ( S , M ) formally as a manifold with tangent spaces T γ C ∞ contr ( S , M ) := Γ( γ ∗ T M ) , choose a formal Riemannian metric on this manifold (i.e. a smoothly varying familyof L inner products on the spaces Γ( γ ∗ T M )) and write down the resulting equationfor the negative gradient flow. A suitable Riemannian metric can be defined bychoosing a smooth S -parametrized family of compatible almost complex structures { J t ∈ J ( M, ω ) } t ∈ S , abbreviated in the following as { J t } , and setting h ξ, η i L := Z S ω ( ξ ( t ) , J t η ( t )) dt for ξ, η ∈ Γ( γ ∗ T M ). Exercise 1.5 then yields the formula d A H ( γ ) η = h J t ( ˙ γ − X t ( γ )) , η i L , so that it seems reasonable to define the so-called unregularized gradient of A H by(1.3) ∇A H ( γ ) := J t ( ˙ γ − X t ( γ )) ∈ Γ( γ ∗ T M ) . Let us also think of a path u : R → C ∞ contr ( S , M ) as a map u : R × S → M , writing u ( s, t ) := u ( s )( t ). The negative gradient flow equation ∂ s u + ∇A H ( u ( s )) = 0 then Chris Wendl
Figure 1.2.
A family of smooth Floer trajectories can degenerateinto a broken Floer trajectory.becomes the elliptic PDE(1.4) ∂ s u + J t ( u ) ( ∂ t u − X t ( u )) = 0 . This is called the
Floer equation , and its solutions are often called
Floer tra-jectories . The relevance of Floer homology to our previous discussion of pseudo-holomorphic curves should now be obvious. Indeed, the resemblance of the Floerequation to the nonlinear Cauchy-Riemann equation is not merely superficial—wewill see in Lecture 6 that the former can always be viewed as a special case of thelatter. In any case, one can use the same set of analytical techniques for both: el-liptic regularity theory implies that Floer trajectories are always smooth, Fredholmtheory and the implicit function theorem imply that (under appropriate assump-tions) they form smooth finite-dimensional moduli spaces. Most importantly, thesame “bubbling off” analysis that underlies Gromov’s compactness theorem can beused to prove that spaces of Floer trajectories are compact up to “breaking”, just asin Morse homology (see Figure 1.2)—this is the main reason for the relation ∂ = 0in Floer homology.We should mention one complication that does not arise either in the study ofclosed holomorphic curves or in finite-dimensional Morse theory. Since the gradientflow in Morse homology takes place on a closed manifold, it is obvious that everygradient flow line asymptotically approaches critical points at both −∞ and + ∞ .The following example shows that in the infinite-dimensional setting of Floer theory,this is no longer true. Example . Consider the Floer equation on M := S = C ∪ {∞} with H := 0and J t defined as the standard complex structure i for every t . Then the orbits of X t are all constant, and a map u : R × S → S satisfies the Floer equation if and onlyif it is holomorphic. Identifying R × S with C ∗ := C \ { } via the biholomorphicmap ( s, t ) e π ( s + it ) , a solution u approaches periodic orbits as s → ±∞ if andonly if the corresponding holomorphic map C ∗ → S extends continuously (andtherefore holomorphically) over 0 and ∞ . But this is not true for every holomorphicmap C ∗ → S , e.g. take any entire function C → C that has an essential singularityat ∞ . ectures on Symplectic Field Theory Exercise . Show that in the above example with an essential singularityat ∞ , the symplectic action A H ( u ( s, · )) is unbounded as s → ∞ . Exercise . Suppose u : R × S → M is a solution to the Floer equation withlim s →±∞ u ( s, · ) = γ ± uniformly for a pair of 1-periodic orbits γ ± ∈ Crit( A H ). Showthat(1.5) A ( γ − ) − A ( γ + ) = Z R × S ω ( ∂ s u, ∂ t u − X t ( u )) ds dt = Z R × S ω ( ∂ s u, J t ( u ) ∂ s u ) ds dt. The right hand side of (1.5) is manifestly nonnegative since J t is compatiblewith ω , and it is strictly positive unless γ − = γ + . It is therefore sensible to callthis expression the energy E ( u ) of a Floer trajectory. The following converse ofExercise 1.9 plays a crucial role in the compactness theory for Floer trajectories, as itguarantees that all the “levels” in a broken Floer trajectory are asymptotically wellbehaved. We will prove a variant of this result in the SFT context (see Prop. 1.23below) in Lecture 9. Proposition . If u : R × S → M is a Floer trajectory with E ( u ) < ∞ andall -periodic orbits of X t are nonegenerate, then there exist orbits γ − , γ + ∈ Crit( A H ) such that lim s →±∞ u ( s, · ) = γ ± uniformly. (cid:3) Remark . It should be emphasized again that we have assumed [ ω ] | π ( M ) =0 throughout this discussion; Floer homology can also be defined under more generalassumptions, but several details become more complicated.For nice comprehensive treatments of Hamiltonian Floer homology—unfortunatelynot always with the same sign conventions as used here—see [ Sal99 , AD14 ]. Notethat this is only one of a few “Floer homologies” that were introduced by Floer inthe late 80’s: the others include
Lagrangian intersection Floer homology [ Flo88a ](which has since evolved into the
Fukaya category , see [
Sei08 ]), and instanton ho-mology [ Flo88c ], an extension of Donaldson’s gauge-theoretic smooth 4-manifoldinvariants to dimension three. The development of new Floer-type theories hassince become a major industry.
A Hamiltonian system on a symplectic manifold (
W, ω ) is called autonomous ifthe Hamiltonian H : W → R does not depend on time. In this case, the Hamiltonianvector field X H defined by ω ( X H , · ) = − dH is time-independent and its orbits are confined to level sets of H . The images ofthese orbits on a given regular level set H − ( c ) depend on the geometry of H − ( c )but not on H itself, as they are the integral curves (also known as characteristics )of the characteristic line field on H − ( c ), defined as the unique direction spannedby a vector X such that ω ( X, Y ) = 0 for all Y tangent to H − ( c ). In 1978, Weinstein[ Wei78 ] and Rabinowitz [
Rab78 ] proved that certain kinds of regular level sets insymplectic manifolds are guaranteed to admit closed characteristics, hence implying Chris Wendl
Figure 1.3.
A star-shaped hypersurface in Euclidean spacethe existence of periodic Hamiltonian orbits. In particular, this is true whenever H − ( c ) is a star-shaped hypersurface in the standard symplectic R n (see Figure 1.3).The following symplectic interpretation of the star-shaped condition providesboth an intuitive reason to believe Rabinowitz’s existence result and motivation forthe more general conjecture of Weinstein. In any symplectic manifold ( W, ω ), a
Liouville vector field is a smooth vector field V that satisfies L V ω = ω. By Cartan’s formula for the Lie derivative, the dual 1-form λ defined by λ := ω ( V, · )satisfies dλ = ω if and only if V is a Liouville vector field; moreover, λ then alsosatisfies L V λ = λ , and it is referred to as a Liouville form . A hypersurface M ⊂ ( W, ω ) is said to be of contact type if it is transverse to a Liouville vectorfield defined on a neighborhood of M . Example . Using coordinates ( q , p , . . . , q n , p n ) on R n , the standard sym-plectic form is written as ω std := n X j =1 dp j ∧ dq j , ectures on Symplectic Field Theory and the Liouville form λ std := P nj =1 ( p j dq j − q j dp j ) is dual to the radial Liouvillevector field V std := 12 n X j =1 (cid:18) p j ∂∂p j + q j ∂∂q j (cid:19) . Any star-shaped hypersurface is therefore of contact type.
Exercise . Suppose (
W, ω ) is a symplectic manifold of dimension 2 n , M ⊂ W is a smoothly embedded and oriented hypersurface, V is a Liouville vector fielddefined near M and λ := ω ( V, · ) is the dual Liouville form. Define a 1-form on M by α := λ | T M .(a) Show that V is positively transverse to M if and only if α satisfies(1.6) α ∧ ( dα ) n − > . (b) If V is positively transverse to M , choose ǫ > − ǫ, ǫ ) × M ֒ → W : ( r, x ) ϕ rV ( x ) , where ϕ tV denotes the time t flow of V . Show thatΦ ∗ λ = e r α, hence Φ ∗ ω = d ( e r α ).The above exercise presents any contact-type hypersurface M ⊂ ( W, ω ) asone member of a smooth 1-parameter family of contact-type hypersurfaces M r := ϕ rV ( M ) ⊂ W , each canonically identified with M such that ω | T M r = e r dα . Inparticular, the characteristic line fields on M r are the same for all r , thus the ex-istence of a closed characteristic on any of these implies that there also exists oneon M . This observation has sometimes been used to prove such existence theorems,e.g. it is used in [ HZ94 , Chapter 4] to reduce Rabinowitz’s result to an “almostexistence” theorem based on symplectic capacities. This discussion hopefully makesthe following conjecture seem believable.
Conjecture . Any closed contact-type hypersurface in a symplectic manifold admits a closed characteristic.
Weinstein’s conjecture admits a natural rephrasing in the language of contactgeometry. A 1-form α on an oriented (2 n − M is called a(positive) contact form if it satisfies (1.6), and the resulting co-oriented hyperplanefield ξ := ker α ⊂ T M is then called a (positive and co-oriented) contact structure . We call the pair(
M, ξ ) a contact manifold , and refer to a diffeomorphism ϕ : M → M ′ as a The adjective “positive” refers to the fact that the orientation of M agrees with the one deter-mined by the volume form α ∧ ( dα ) n − ; we call α a negative contact form if these two orientationsdisagree. It is also possible in general to define contact structures without co-orientations, but con-tact structures of this type will never appear in these notes; for our purposes, the co-orientation is always considered to be part of the data of a contact structure. Chris Wendl contactomorphism from (
M, ξ ) to ( M ′ , ξ ′ ) if ϕ ∗ maps ξ to ξ ′ and also preservesthe respective co-orientations. Equivalently, if ξ and ξ ′ are defined via contact forms α and α ′ respectively, this means ϕ ∗ α ′ = f α for some f ∈ C ∞ ( M, (0 , ∞ )) . Contact topology studies the category of contact manifolds (
M, ξ ) up to con-tactomorphism. The following basic result provides one good reason to regard ξ rather than α as the geometrically meaningful data, as the result holds for contact structures , but not for contact forms . Theorem . If M is a closed (2 n − -dimensionalmanifold and { ξ t } t ∈ [0 , is a smooth -parameter family of contact structures on M ,then there exists a smooth -parameter family of diffeomorphisms { ϕ t } t ∈ [0 , suchthat ϕ = Id and ( ϕ t ) ∗ ξ = ξ t . Proof.
See [
Gei08 , § Wend , Theorem 1.6.12]. (cid:3)
A corollary is that while the contact form α induced on a contact-type hyper-surface M ⊂ ( W, ω ) via Exercise 1.13 is not unique, its induced contact structure isunique up to isotopy. Indeed, the space of all Liouville vector fields transverse to M is very large (e.g. one can add to V any sufficiently small Hamiltonian vector field),but it is convex , hence any two choices of the induced contact form α on M areconnected by a smooth 1-parameter family of contact forms, implying an isotopy ofcontact structures via Gray’s theorem. Exercise . If α is a nowhere zero 1-form on M and ξ = ker α , show that α is contact if and only if dα | ξ defines a symplectic vector bundle structure on ξ → M .Moreover, the orientation of ξ determined by this symplectic bundle structure iscompatible with the co-orientation determined by α and the orientation of M forwhich α ∧ ( dα ) n − > dα | ξ is nondegeneratewhen α is contact, ker dα ⊂ T M is always 1-dimensional and transverse to ξ . Definition . Given a contact form α on M , the Reeb vector field is theunique vector field R α that satisfies dα ( R α , · ) ≡ , and α ( R α ) ≡ . Exercise . Show that the flow of any Reeb vector field R α preserves both ξ = ker α and the symplectic vector bundle structure dα | ξ . Conjecture . On any closed con-tact manifold ( M, ξ ) with contact form α , the Reeb vector field R α admits a periodicorbit. To see that this is equivalent to the symplectic version of the conjecture, ob-serve that any contact manifold (
M, ξ = ker α ) can be viewed as the contact-typehypersurface { } × M in the open symplectic manifold( R × M, d ( e r α )) , called the symplectization of ( M, ξ ). ectures on Symplectic Field Theory Exercise . Recall that on any smooth manifold M , there is a tautological1-form λ that locally takes the form λ = P nj =1 p j dq j in any choice of local coordi-nates ( q , . . . , q n ) on a neighbood U ⊂ M , with ( p , . . . , p n ) denoting the inducedcoordinates on the cotangent fibers over U . This is a Liouville form, with dλ defin-ing the canonical symplectic structure of T ∗ M . Now if ξ ⊂ T M is a co-orientedhyperplane field on M , consider the submanifold S ξ M := (cid:8) p ∈ T ∗ M (cid:12)(cid:12) ker p = ξ and p ( X ) > X ∈ T M pos. transverse to ξ (cid:9) . Show that ξ is contact if and only if S ξ M is a symplectic submanifold of ( T ∗ M, dλ ),and the Liouville vector field on T ∗ M dual to λ is tangent to S ξ M . Moreover, if ξ iscontact, then any choice of contact form for ξ determines a diffeomorphism of S ξ M to R × M identifying the Liouville form λ along S ξ M with e r α . Remark . Exercise 1.20 shows that up to symplectomorphism, our defi-nition of the symplectization of (
M, ξ ) above actually depends only on ξ and noton α .In 1993, Hofer [ Hof93 ] introduced a new approach to the Weinstein conjecturethat was based in part on ideas of Gromov and Floer. Fix a contact manifold (
M, ξ )with contact form α , and let J ( α ) ⊂ J ( R × M )denote the nonempty and contractible space of all almost complex structures J on R × M satisfying the following conditions:(1) The natural translation action on R × M preserves J ;(2) J ∂ r = R α and J R α = − ∂ r , where r denotes the canonical coordinate onthe R -factor in R × M ;(3) J ξ = ξ and dα ( · , J · ) | ξ defines a bundle metric on ξ .It is easy to check that any J ∈ J ( α ) is compatible with the symplectic structure d ( e r α ) on R × M . Moreover, if γ : R → M is any periodic orbit of R α with period T >
0, then for any J ∈ J ( α ), the so-called trivial cylinder u : R × S → R × M : ( s, t ) ( T s, γ ( T t ))is a J -holomorphic curve. Following Floer, one version of Hofer’s idea would be tolook for J -holomorphic cylinders that satisfy a finite energy condition as in Prop. 1.10forcing them to approach trivial cylinders asymptotically—the existence of such acylinder would then imply the existence of a closed Reeb orbit and thus prove theWeinstein conjecture. The first hindrance is that the “obvious” definition of energyin this context, Z R × S u ∗ d ( e r α ) , is not the right one: this integral is infinite if u is a trivial cylinder. To circumventthis, notice that every J ∈ J ( α ) is also compatible with any symplectic structureof the form ω ϕ := d ( e ϕ ( r ) α ) , Chris Wendl where ϕ is a function chosen freely from the set(1.7) T := (cid:8) ϕ ∈ C ∞ ( R , ( − , (cid:12)(cid:12) ϕ ′ > (cid:9) . Essentially, choosing ω ϕ means identifying R × M with a subset of the boundedregion ( − , × M , in which trivial cylinders have finite symplectic area. Sincethere is no preferred choice for the function ϕ , we define the Hofer energy of a J -holomorphic curve u : Σ → R × M by(1.8) E ( u ) := sup ϕ ∈T Z Σ u ∗ ω ϕ . This has the desired property of being finite for trivial cylinders, and it is alsononnegative, with strict positivity whenever u is not constant.Another useful observation from [ Hof93 ] was that if the goal is to find periodicorbits, then we need not restrict our attention to J -holomorphic cylinders in par-ticular. One can more generally consider curves defined on an arbitrary punctured Riemann surface ˙Σ := Σ \ Γ , where (Σ , j ) is a closed connected Riemann surface and Γ ⊂ Σ is a finite set ofpunctures. For any ζ ∈ Γ, one can find coordinates identifying some puncturedneighborhood of ζ biholomorphically with the closed punctured disk˙ D := D \ { } ⊂ C , and then identify this with either the positive or negative half-cylinder Z + := [0 , ∞ ) × S , Z − := ( −∞ , × S via the biholomorphic maps Z + → ˙ D : ( s, t ) e − π ( s + it ) , Z − → ˙ D : ( s, t ) e π ( s + it ) . We will refer to such a choice as a (positive or negative) holomorphic cylindricalcoordinate system near ζ , and in this way, we can present ( ˙Σ , j ) as a Riemannsurface with cylindrical ends , i.e. the union of some compact Riemann surface withboundary with a finite collection of half-cylinders Z ± on which j takes the standardform j∂ s = ∂ t . Note that the standard cylinder R × S is a special case of this, asit can be identified biholomorphically with S \ { , ∞} . Another important specialcase is the plane, C = S \ {∞} .If u : ( ˙Σ , j ) → ( R × M, J ) is a J -holomorphic curve and ζ ∈ Γ is one of itspunctures, we will say that u is positively/negatively asymptotic to a T -periodicReeb orbit γ : R → M at ζ if one can choose holomorphic cylindrical coordinates( s, t ) ∈ Z ± near ζ such that u ( s, t ) = exp ( T s,γ ( T t )) h ( s, t ) for | s | sufficiently large , Strictly speaking, the energy defined in (1.8) is not identical to the notion introduced in[
Hof93 ] and used in many of Hofer’s papers, but it is equivalent to it in the sense that uniformbounds on either notion of energy imply uniform bounds on the other. ectures on Symplectic Field Theory ˙Σ u {∞} × M {−∞} × M Figure 1.4.
An asymptotically cylindrical holomorphic curve in asymplectization, with genus 1, one positive puncture and two negativepunctures.where h ( s, t ) is a vector field along the trivial cylinder satisfying h ( s, · ) → | s | → ∞ , and the exponential map is defined with respect to any R -invariant choice of Riemannian metric on R × M . We say that u : ( ˙Σ , j ) → ( R × M, J )is asymptotically cylindrical if it is (positively or negatively) asymptotic to someclosd Reeb orbit at each of its punctures. Note that this partitions the finite set ofpunctures Γ ⊂ Σ into two subsets, Γ = Γ + ∪ Γ − , the positive and negative punctures respectively, see Figure 1.4. Exercise . Suppose u : ( ˙Σ , j ) → ( R × M, J ) is an asymptotically cylindrical J -holomorphic curve, with the asymptotic orbit at each puncture ζ ∈ Γ ± denotedby γ ζ , having period T ζ >
0. Show that X ζ ∈ Γ + T ζ − X ζ ∈ Γ − T ζ = Z ˙Σ u ∗ dα ≥ , with equality if and only if the image of u is contained in that of a trivial cylinder.In particular, u must have at least one positive puncture unless it is constant. Showalso that E ( u ) is finite and satisfies an upper bound determined only by the periodsof the positive asymptotic orbits.The following analogue of Prop. 1.10 will be proved in Lecture 9. For simplicity,we shall state a weakened version of what Hofer proved in [ Hof93 ], which did notrequire any nondegeneracy assumption. A T -periodic Reeb orbit γ : R → M iscalled nondegenerate if the Reeb flow ϕ tα has the property that its linearizationalong the contact bundle (cf. Exercise 1.18), dϕ Tα ( γ (0)) | ξ γ (0) : ξ γ (0) → ξ γ (0) does not have 1 as an eigenvalue. Note that since R α is not time-dependent, closedReeb orbits are never completely isolated—they always exist in S -parametrizedfamilies—but these families are isolated in the nondegenerate case. Chris Wendl
Proposition . Suppose ( M, ξ ) is a closed contact manifold, with a contactform α such that all closed Reeb orbits are nondegenerate. If u : ( ˙Σ , j ) → ( R × M, J ) is a J -holomorphic curve with E ( u ) < ∞ on a punctured Riemann surface such thatnone of the punctures are removable, then u is asymptotically cylindrical. (cid:3) The main results in [
Hof93 ] state that under certain assumptions on a closedcontact 3-manifold (
M, ξ ), namely if either ξ is overtwisted (as defined in [ Eli89 ])or π ( M ) = 0, one can find for any contact form α on ( M, ξ ) and any J ∈ J ( α ) afinite-energy J -holomorphic plane. By Proposition 1.23, this implies the existenceof a contractible periodic Reeb orbit and thus proves the Weinstein conjecture inthese settings. After the developments described in the previous three sections, it seemed nat-ural that one might define invariants of contact manifolds via a Floer-type theorygenerated by closed Reeb orbits and counting asymptotically cylindrical holomor-phic curves in symplectizations. This theory is what is now called SFT, and itsbasic structure was outlined in a paper by Eliashberg, Givental and Hofer [
EGH00 ]in 2000, though some of its analytical foundations remain unfinished in 2016. Theterm “field theory” is an allusion to “topological quantum field theories,” whichassociate vector spaces to certain geometric objects and morphisms to cobordismsbetween those objects. Thus in order to place SFT in its proper setting, we need tointroduce symplectic cobordisms between contact manifolds.Recall that if M + and M − are smooth oriented closed manifolds of the samedimension, an oriented cobordism from M − to M + is a compact smooth orientedmanifold W with oriented boundary ∂W = − M − ⊔ M + , where − M − denotes M − with its orientation reversed. Given positive contact struc-tures ξ ± on M ± , we say that a symplectic manifold ( W, ω ) is a symplectic cobor-dism from ( M − , ξ − ) to ( M + , ξ + ) if W is an oriented cobordism from M − to M + such that both components of ∂W are contact-type hypersurfaces with induced con-tact structures isotopic to ξ ± . Note that our chosen orientation conventions implyin this case that the Liouville vector field chosen near ∂W must point outward at M + and inward at M − ; we say in this case that M + is a symplectically convex boundary component, while M − is symplectically concave . As important specialcases, ( W, ω ) is a symplectic filling of ( M + , ξ + ) if M − = ∅ , and it is a symplecticcap of ( M − , ξ − ) if M + = ∅ . In the literature, fillings and caps are sometimes alsoreferred to as convex fillings or concave fillings respectively.The contact-type condition implies the existence of a Liouville form λ near ∂W with dλ = ω , such that by Exercise 1.13, neighborhoods of M + and M − in W canbe identified with the collars (see Figure 1.5)( − ǫ, × M + or [0 , ǫ ) × M − We assume of course that W is assigned the orientation determined by its symplectic form. ectures on Symplectic Field Theory (( − ǫ, × M + , d ( e r α + ))([0 , ǫ ) × M − , d ( e r α − ))( W, ω ) Figure 1.5.
A symplectic cobordism with concave boundary( M − , ξ − ) and convex boundary ( M + , ξ + ), with symplectic collar neigh-borhoods defined by flowing along Liouville vector fields near theboundary.respectively for sufficiently small ǫ >
0, with λ taking the form λ = e r α ± , where α ± := λ | T M ± are contact forms for ξ ± . The symplectic completion of( W, ω ) is the noncompact symplectic manifold ( c W , ˆ ω ) defined by attaching cylindri-cal ends to these collar neighborhoods (Figure 1.6):( c W , ˆ ω ) = (( −∞ , × M − , d ( e r α − )) ∪ M − ( W, ω ) ∪ M + ([0 , ∞ ) × M + , d ( e r α + )) . (1.9)In this context, the symplectization ( R × M, d ( e r α )) is symplectomorphic to thecompletion of the trivial symplectic cobordism ([0 , × M, d ( e r α )) from ( M, ξ =ker α ) to itself. More generally, the object in the following easy exercise can alsosensibly be called a trivial symplectic cobordism: Exercise . Suppose (
M, ξ ) is a closed contact manifold with contact form α , and f ± : M → R is a pair of functions with f − < f + everywhere. Show that thedomain (cid:8) ( r, x ) ∈ R × M (cid:12)(cid:12) f − ( x ) ≤ r ≤ f + ( x ) (cid:9) ⊂ R × M defines a symplectic cobordism from ( M, ξ ) to itself, with a global Liouville form λ = e r α inducing contact forms e f − α and e f + α on its concave and convex boundariesrespectively.We say that ( W, ω ) is an exact symplectic cobordism or Liouville cobor-dism if the Liouville form λ can be extended from a neighborhood of ∂W to definea global primitive of ω on W . Equivalently, this means that ω admits a global Li-ouville vector field that points inward at M − and outward at M + . An exact filling of ( M + , ξ + ) is an exact cobordism whose concave boundary is empty. Observe thatif ( W, ω ) is exact, then its completion ( c W , ˆ ω ) also inherits a global Liouville form. Exercise . Use Stokes’ theorem to show that there is no such thing as anexact symplectic cap. Chris Wendl − ( W, ω )(( − ǫ, × M + , d ( e r α + ))([0 , ǫ ) × M − , d ( e r α − ))([0 , ∞ ) × M + , d ( e r α + ))(( −∞ , × M − , d ( e r α − )) Figure 1.6.
The completion of a symplectic cobordismThe above exercise hints at an important difference between cobordisms in the symplectic as opposed to the oriented smooth category: symplectic cobordisms arenot generally reversible. If W is an oriented cobordism from M − to M + , thenreversing the orientation of W produces an oriented cobordism from M + to M − .But one cannot simply reverse orientations in the symplectic category, since theorientation is determined by the symplectic form. For example, many obstructionsto the existence of symplectic fillings of given contact manifolds are known—someof them defined in terms of SFT—but we do not know any obstructions at all tosymplectic caps, in fact it is known that all contact 3-manifolds admit them.The definitions for holomorphic curves in symplectizations in the previous sec-tion generalize to completions of symplectic cobordisms in a fairly straightforwardway since these completions look exactly like symplectizations outside of a compactsubset. Define J ( W, ω, α + , α − ) ⊂ J ( c W )as the space of all almost complex structures J on c W such that J | W ∈ J ( W, ω ) , J | [0 , ∞ ) × M + ∈ J ( α + ) and J | ( −∞ , × M − ∈ J ( α − ) . ectures on Symplectic Field Theory ˙Σ u c Wd ( Figure 1.7.
An asymptotically cylindrical holomorphic curve in acompleted symplectic cobordism, with genus 2, one positive punctureand two negative punctures.Occasionally it is useful to relax the compatibility condition on W to tameness, i.e. J | W ∈ J τ ( W, ω ), producing a space that we shall denote by J τ ( W, ω, α + , α − ) ⊂ J ( c W ) . As in Prop. 1.2, both of these spaces are nonempty and contractible. We can thenconsider asymptotically cylindrical J -holomorphic curves u : ( ˙Σ = Σ \ (Γ + ∪ Γ − ) , j ) → ( c W , J ) , which are proper maps asymptotic to closed orbits of R α ± in M ± at punctures in Γ ± ,see Figure 1.7.One must again tinker with the symplectic form on c W in order to define a notionof energy that is finite when we need it to be. We generalize (1.7) as T := (cid:8) ϕ ∈ C ∞ ( R , ( − , (cid:12)(cid:12) ϕ ′ > ϕ ( r ) = r near r = 0 (cid:9) , and associate to each ϕ ∈ T a symplectic form ˆ ω ϕ on c W defined byˆ ω ϕ := d ( e ϕ ( r ) α + ) on [0 , ∞ ) × M + ,ω on W ,d ( e ϕ ( r ) α − ) on ( −∞ , × M − . One can again check that every J ∈ J ( W, ω, α + , α − ) or J τ ( W, ω, α + , α − ) is com-patible with or, respectively, tamed by ˆ ω ϕ for every ϕ ∈ T . Thus it makes sense to It seems natural to wonder whether one could not also relax the conditions on the cylindricalends and require J | ξ ± to be tamed by dα ± | ξ ± instead of compatible with it. I do not currentlyknow whether this works, but in later lectures we will see some reasons to worry that it might not. Chris Wendl define the energy of u : ( ˙Σ , j ) → ( c W , J ) by E ( u ) := sup ϕ ∈T Z ˙Σ u ∗ ˆ ω ϕ . It will be a straightforward matter to generalize Proposition 1.23 and show thatfinite energy implies asymptotically cylindrical behavior in completed cobordisms.
Exercise . Show that if (
W, ω ) is an exact cobordism, then every asymp-totically cylindrical J -holomorphic curve in c W has at least one positive puncture. We can now sketch the algebraic structure of SFT. We shall ignore or suppressseveral pesky details that are best dealt with later, some of them algebraic, othersanalytical. Due to analytical problems, some of the “theorems” that we shall (oftenimprecisely) state in this section are not yet provable at the current level of tech-nology, though we expect that they will be soon. We shall use quotation marks toindicate this caveat wherever appropriate.The standard versions of SFT all define homology theories with varying levels ofalgebraic structure which are meant to be invariants of a contact manifold (
M, ξ ).The chain complexes always depend on certain auxiliary choices, including a nonde-generate contact form α and a generic J ∈ J ( α ). The generators consist of formalvariables q γ , one for each closed Reeb orbit γ . In the most straightforward gen-eralization of Hamiltonian Floer homology, the chain complex is simply a graded Q -vector space generated by the variables q γ , and the boundary map is defined by ∂ CCH q γ = X γ ′ (cid:0) M ( γ, γ ′ ) (cid:14) R (cid:1) q γ ′ , where M ( γ, γ ′ ) is the moduli space of J -holomorphic cylinders in R × M with apositive puncture asymptotic to γ and a negative puncture asymptotic to γ ′ , and thesum ranges over all orbits γ ′ for which this moduli space is 1-dimensional. The count M ( γ, γ ′ ) / R ) is rational, as it includes rational weighting factors that depend oncombinatorial information and are best not discussed right now. “Theorem” . If α admits no contractible Reeb orbits, then ∂ = 0 , andthe resulting homology is independent of the choices of α with this property andgeneric J ∈ J ( α ) . The invariant arising from this result is known as cylindrical contact homol-ogy , and it is sometimes quite easy to work with when it is well defined, though ithas the disadvantage of not always being defined. Namely, the relation ∂ = 0can fail if α admits contractible Reeb orbits, because unlike in Floer homology, thecompactification of the space of cylinders M ( γ, γ ′ ) generally includes objects thatare not broken cylinders. In fact, the objects arising in the “SFT compactification” Actually I should be making a distinction here between “good” and “bad” Reeb orbits, butlet’s discuss that later; see Lecture 11. Similar combinatorial factors are hidden behind the symbol “ ∂ CH and H , and will be discussed in earnest in Lecture 12. ectures on Symplectic Field Theory c W c Wu k ( M + , ξ + )( M − , ξ − ) v +1 v v − v − v − R × M + R × M − R × M − R × M − Figure 1.8.
Degeneration of a sequence u k of finite energy punc-tured holomorphic curves with genus 2, one positive puncture and twonegative punctures in a symplectic cobordism. The limiting holomor-phic building ( v +1 , v , v − , v − , v − ) in this example has one upper levelliving in the symplectization R × M + , a main level living in c W , andthree lower levels, each of which is a (possibly disconnected) finite-energy punctured nodal holomorphic curve in R × M − . The buildinghas arithmetic genus 2 and the same numbers of positive and negativepunctures as u k .of moduli spaces of finite-energy curves in completed cobordisms can be quite elab-orate, see Figure 1.8. The combinatorics of the situation are not so bad howeverif the cobordism is exact, as is the case for a symplectization: Exercise 1.26 thenprevents curves without positive ends from appearing. The only possible degen-erations for cylinders then consist of broken configurations whose levels each have exactly one positive puncture and arbitrary negative punctures; moreover, all butone of the negative punctures must eventually be capped off by planes, which is why“Theorem” 1.27 holds in the absence of planes.If planes do exist, then one can account for them by defining the chain complexas an algebra rather than a vector space, producing the theory known as contacthomology . For this, the chain complex is taken to be a graded unital algebra over Chris Wendl Q , and we define ∂ CH q γ = X ( γ ,...,γ m ) (cid:0) M ( γ ; γ , . . . , γ m ) (cid:14) R (cid:1) q γ . . . q γ m , with M ( γ ; γ , . . . , γ m ) denoting the moduli space of punctured J -holomorphic spheresin R × M with a positive puncture at γ and m negative punctures at the orbits γ , . . . , γ m , and the sum ranges over all integers m ≥ m -tuples of orbits forwhich the moduli space is 1-dimensional. The action of ∂ CH is then extended to thewhole algebra via a graded Leibniz rule ∂ CH ( q γ q γ ′ ) := ( ∂ CH q γ ) q γ ′ + ( − | γ | q γ ( ∂ CH q γ ′ ) . The general compactness and gluing theory for genus zero curves with one positivepuncture now implies: “Theorem” . ∂ = 0 , and the resulting homology is (as a graded unital Q -algebra) independent of the choices α and J . Maybe you’ve noticed the pattern: in order to accommodate more general classesof holomorphic curves, we need to add more algebraic structure. The full SFT algebra counts all rigid holomorphic curves in R × M , including all combinations ofpositive and negative punctures and all genera. Here is a brief picture of what itlooks like. Counting all the 1-dimensional moduli spaces of J -holomorphic curvesmodulo R -translation in R × M produces a formal power series H := X (cid:16) M g ( γ +1 , . . . , γ + m + ; γ − , . . . , γ − m − ) . R (cid:17) q γ − . . . q γ − m − p γ +1 . . . p γ + m + ~ g − , where the sum ranges over all integers g, m + , m − ≥ ~ and p γ (one for each orbit γ ) are additional formal variables, and M g ( γ +1 , . . . , γ + m + ; γ − , . . . , γ − m − )denotes the moduli space of J -holomorphic curves in R × M with genus g , m + positive punctures at the orbits γ +1 , . . . , γ + m + , and m − negative punctures at theorbits γ − , . . . , γ + m − . We can regard H as an operator on a graded algebra W offormal power series in the variables { p γ } , { q γ } and ~ , equipped with a graded bracketoperation that satisfies the quantum mechanical commutation relation[ p γ , q γ ] = κ γ ~ , where κ γ is a combinatorial factor that is best ignored for now. Note that due to thesigns that accompany the grading, odd elements F ∈ W need not satisfy [ F , F ] = 0,and H itself is an odd element, thus the following statement is nontrivial; in fact,it is the algebraic manifestation of the general compactness and gluing theory forpunctured holomorphic curves in symplectizations. “Theorem” . [ H , H ] = 0 , hence by the graded Jacobi identity, H deter-mines an operator D SFT : W → W : F [ H , F ] satisfying D = 0 . The resulting homology depends on ( M, ξ ) but not on theauxiliary choices α and J . ectures on Symplectic Field Theory It takes some time to understand how pictures such as Figure 1.8 translateinto algebraic relations like [ H , H ] = 0, but this is a subject we’ll come back to.There is also an intermediate theory between contact homology and full SFT, called rational SFT , which counts only genus zero curves with arbitrary positive andnegative punctures. Algebraically, it is obtained from the full SFT algebra as a“semiclassical approximation” by discarding higher-order factors of ~ so that thecommutation bracket in W becomes a graded Poisson bracket. We will discuss allof this in Lecture 12. We briefly mention two applications that we will be able to establish rigorouslyusing the methods developed in this book. Since SFT itself is not yet well definedin full generality, this sometimes means using SFT for inspiration while provingcorollaries via more direct methods. T . The 3-torus T = S × S × S withcoordinates ( t, θ, φ ) admits a sequence of contact structures ξ k := ker (cos(2 πkt ) dθ + sin(2 πkt ) dφ ) , one for each k ∈ N . These cannot be distinguished from each other by any classicalinvariants, e.g. they all have the same Euler class, in fact they are all homotopic asco-oriented 2-plane fields. Nonetheless: Theorem . For k = ℓ , ( T , ξ k ) and ( T , ξ ℓ ) are not contactomorphic. We will be able to prove this in Lecture 10 by rigorously defining and computingcylindrical contact homology for a suitable choice of contact forms on ( T , ξ k ). Consider a closed connectedand oriented surface Σ presented as Σ + ∪ Γ Σ − , where Σ ± ⊂ Σ are each (not neces-sarily connected) compact surfaces with a common boundary Γ. By an old result ofLutz [
Lut77 ], the 3-manifold S × Σ admits a unique isotopy class of S -invariantcontact structures ξ Γ such that the loops S × { z } are positively/negatively trans-verse to ξ Γ for z ∈ ˚Σ ± and tangent to ξ Γ for z ∈ Γ. Now for each k ∈ N , define( V k , ξ k ) := ( S × Σ , ξ Γ )where Σ = Σ + ∪ Γ Σ − is chosen such that Γ has k connected components, Σ − isconnected with genus zero, and Σ + is connected with positive genus (see Figure 1.9). Theorem . The contact manifolds ( V k , ξ k ) do not admit any symplecticfillings. Moreover, if k > ℓ , then there exists no exact symplectic cobordism from ( V k , ξ k ) to ( V ℓ , ξ ℓ ) . For these examples, one can use explicit constructions from [
Wen13 , Avd ] toshow that non-exact cobordisms from ( V k , ξ k ) to ( V ℓ , ξ ℓ ) do exist, and so do exactcobordisms from ( V ℓ , ξ ℓ ) to ( V k , ξ k ), thus both the directionality of the cobordismrelation and the distinction between exact and non-exact are crucial. The proofof the theorem, due to the author with Latschev and Hutchings [ LW11 ], uses a Chris Wendl − S × S × ( W, dλ ) ( V , ξ )( V , ξ ) Figure 1.9.
This exact symplectic cobordism does not exist.numerical contact invariant based on the full SFT algebra—in particular, the curvesthat cause this phenomenon have multiple positive ends and are thus not seen bycontact homology. We will introduce the relevant numerical invariant in Lecture 13and compute it for these examples in Lecture 16.ECTURE 2
Basics on holomorphic curvesContents
In this lecture we begin studying the analysis of J -holomorphic curves. Thecoverage will necessarily be a bit sparse in some places, but more detailed proofs ofeverything in this lecture can be found in [ Wend ]. In order to motivate the study of linear Cauchy-Riemann type operators, webegin with a formal discussion of the nonlinear Cauchy-Riemann equation and itslinearization.Fix a Riemann surface (Σ , j ) and almost complex manifold (
W, J ), and supposethat we wish to understand the structure of some space of the form(2.1) { u : Σ → W | T u ◦ j = J ◦ T u plus further conditions } , where the “further conditions” (which we will for now leave unspecified) may imposeconstraints on e.g. the regularity of u , as well as its boundary and/or asymptoticbehavior. The standard approach in global analysis can be summarized as follows: Step 1:
Construct a smooth Banach manifold B of maps u : Σ → W such that allthe solutions we’re interested in will be elements of B . The tangent spaces T u B are then Banach spaces of sections of u ∗ T W . Step 2:
Construct a smooth Banach space bundle
E → B such that for each u ∈ B ,the fiber E u is a Banach space of sections of the vector bundleHom C ( T Σ , u ∗ T W ) → Σof complex-antilinear bundle maps ( T Σ , j ) → ( u ∗ T W, J ). Since our purposeis to study a first-order PDE, we need the sections in E u to be “one stepless regular” than the maps in B , e.g. if B consists of maps of Sobolev class W k,p , then the sections in E u should be of class W k − ,p . Chris Wendl
Step 3:
Show that ¯ ∂ J : B → E : u T u + J ( u ) ◦ T u ◦ j defines a smooth section of E → B , whose zero set is precisely the space ofsolutions (2.1).
Step 4:
Show that under suitable assumptions (e.g. on regularity and asymptoticbehavior), one can arrange such that for every u ∈ ¯ ∂ − J (0), the lineariza-tion of ¯ ∂ J , D ¯ ∂ J ( u ) : T u B → E u is a Fredholm operator and is generically surjective. (In geometric terms,this would mean that ¯ ∂ J is transverse to the zero section .) Step 5:
Using the implicit function theorem in Banach spaces (see [
Lan93 ]), thesurjectivity of D ¯ ∂ J ( u ) implies that ¯ ∂ − J (0) is a smooth finite-dimensionalmanifold, with its tangent space at each u ∈ ¯ ∂ − J (0) canonically identifiedwith ker D ¯ ∂ J ( u ), hence the dimension of ¯ ∂ − J (0) near u equals the Fredholmindex of D ¯ ∂ J ( u ).Without worrying about the fact that these are actually not Banach spaces, andsome Sobolev completion is needed, let us assume, for simplicity, that the bundle E → B has as base the space B = C ∞ (Σ , W ) and the fiber over u ∈ B is given by E u = Hom C ( T Σ , u ∗ T W ). The linearization of the section ∂ J at a point u ∈ ¯ ∂ − J (0)should then take the form D u : Γ( u ∗ T W ) → Ω , (Σ , u ∗ T W ) , where the right hand side denotes the space of u ∗ T W -valued (0 , C ( T Σ , u ∗ T W ) = T , Σ ⊗ C u ∗ T W , where T , Σdenotes the (0 , E . Choose a connection ∇ on W , and recall the fact that thisnaturally induces a connection on the bundles T , Σ ⊗ C u ∗ T W and End( u ∗ T W )by setting ∇ ( α ⊗ s ) = α ⊗ ∇ s and ( ∇ J ) s = ∇ ( J s ) − J ∇ s , for s ∈ Γ( u ∗ T W ), J ∈ End( u ∗ T W ) and α ∈ Γ( T , Σ). We shall make the ansatz that for any smooth1-parameter family of maps u ρ : Σ → W for ρ ∈ ( − ǫ, ǫ ) and a section η ρ ∈ E u ρ along the path (i.e a section of the pullback bundle of E under the map ( − ǫ, ǫ ) → B mapping ρ to u ρ ), the connection takes the form( ∇ ρ η ρ ) X = ∇ ρ ( η ρ ( X )) , for X ∈ T Σ, where this expression should be interpreted as the pullback connectionunder the map displayed above. The tensorial property of connections implies that ∇ ρ η ρ does not depend on the connection at the values ρ for which η ρ = 0.Given u ∈ ∂ − J (0) and η in T u B = Γ( u ∗ T W ), take a one-parameter family u ρ ∈ B with u = u and ∂ ρ u ρ | ρ =0 = η . We then have that D u η = ∇ ρ (cid:0) ¯ ∂ J ( u ρ ) (cid:1)(cid:12)(cid:12) ρ =0 = ∇ ρ ( T u ρ + J ( u ρ ) ◦ T u ρ ◦ j ) | ρ =0 . Since ¯ ∂ J u = 0, this is independent of the connection, and we may therefore choose ∇ to be symmetric. ectures on Symplectic Field Theory Fix a point z ∈ Σ and choose local holomorphic coordinates s + it around it. Thesymmetry of the connection implies ∇ ρ ∂ s u ρ | ρ =0 = ∇ s ∂ ρ u ρ | ρ =0 = ∇ s η , and similarlyfor the variable t . Observing also that ∇ η J = ∇ ∂ ρ u ρ | ρ =0 J = ∇ ρ ( J ( u ρ )) | ρ =0 , andusing the above ansatz, we obtain( D u η ) ∂ s = ∇ ρ ( ∂ s u ρ + J ( u ρ ) ◦ ∂ t u ρ ) | ρ =0 = ∇ s η + J ( u ) ∇ t η + ( ∇ η J ) ∂ t u Since D u η is an antilinear map, and ∂ t = j∂ s , it is therefore determined by itsaction on ∂ s . One can check that the operator on the right hand side below is alsoantilinear, and thus removing the ∂ s , we obtain(2.2) D u η = ∇ η + J ( u ) ◦ ∇ η ◦ j + ( ∇ η J ) ◦ T u ◦ j. Definition . Fix a complex vector bundle E over a Riemann surface (Σ , j ).A (real) linear Cauchy-Riemann type operator on E is a real-linear first-orderdifferential operator D : Γ( E ) → Ω , (Σ , E )such that for every f ∈ C ∞ (Σ , R ) and η ∈ Γ( E ),(2.3) D ( f η ) = ( ¯ ∂f ) η + f D η, where ¯ ∂f denotes the complex-valued (0 , df + i df ◦ j .Observe that D is complex linear if and only if the Leibniz rule (2.3) also holdsfor all smooth complex-valued functions f , not just real-valued. It is a standardresult in complex geometry that choosing a complex-linear Cauchy-Riemann typeoperator D on E is equivalent to endowing it with the structure of a holomorphic vector bundle, where local sections η are defined to be holomorphic if and onlyif D η = 0. Indeed, every holomorphic bundle comes with a canonical Cauchy-Riemann operator that is expressed as ¯ ∂ in holomorphic trivializations, and in theother direction, the equivalence follows from a local existence result for solutions tothe equation D η = 0, proved in § Exercise . If D is a linear Cauchy-Riemann type operator on E , prove thatevery other such operator is of the form D + A where A : E → Hom C ( T Σ , E ) isa smooth linear bundle map. Using this, show that in suitable local trivializationsover a subset U ⊂
Σ identified biholomorphically with an open set in C , everyCauchy-Riemann type operator D takes the form D = ¯ ∂ + A : C ∞ ( U , C m ) → C ∞ ( U , C m ) , where ¯ ∂ = ∂ s + i∂ t in complex coordinates z = s + it and A ∈ C ∞ ( U , End R ( C m )). Exercise . Verify that the linearized operator D u of (2.2) is a real-linearCauchy-Riemann type operator. This statement about the existence of holomorphic vector bundle structures is true whenthe base is a Riemann surface, but not if it is a higher-dimensional complex manifold. In higherdimensions there are obstructions, see e.g. [
Kob87 ]. Chris Wendl
In this section, we review a few general properties of Sobolev spaces that areessential for applications in nonlinear analysis. The results stated here are explainedin more detail in Appendix A.Throughout this section we consider functions with values in C unless otherwisespecified, and defined on an open domain U in either R n or a quotient of R n onwhich the Lebesgue measure is well defined. Certain regularity assumptions mustgenerally be placed on the boundary of U in order for all the results stated belowto hold; we will ignore this detail except to mention that the necessary assumptionsare satisfied for the two classes of domains that we are most interested in, which are U = ˚ D ⊂ C , U = (0 , L ) × S ⊂ C / Z , < L ≤ ∞ . Here D denotes the closed unit disk and ˚ D is its interior. Certain results will bespecified to hold only for bounded domains, which means in practice that they holdon ˚ D and (0 , L ) × S for any L >
0, but not on (0 , ∞ ) × S .Recall that for p ∈ [1 , ∞ ) we define the L p norm of a measurable function f : U → R m to be k f k L p = (cid:18)Z U | f | p (cid:19) /p . For the space L ∞ we define the norm to be the essential supremum of f over U .Denote by C ∞ ( U ) ⊂ C ∞ ( U )the set of smooth functions with compact support in U . We say a function f hasa weak j -th partial derivative g if the integration by parts formula holds for all ϕ ∈ C ∞ ( U ): Z U gϕ = − Z U f ∂ j ϕ. Equivalently, this means that g is a partial derivative of f in the sense of distribu-tions (see e.g. [ LL01 ]). Higher order weak partial derivatives are defined similarly:recall that for a multiindex α = ( i , ...i n ) we denote ∂ α f = ∂ | α | f∂x i . . . ∂x i n n , where | α | := P j i j . We then write ∂ α f = g if for all ϕ ∈ C ∞ ( U ), Z U gϕ = ( − | α | Z U f ∂ α ϕ. Now we may define W k,p ( U ) to be the set of functions on U with weak partialderivatives up to order k lying in L p , and define the norm of such a function by: k f k W k,p = X | α |≤ k k ∂ α f k L p . ectures on Symplectic Field Theory As W k,p ( U ) can be regarded as a subset of a k -fold product of L p ( U ), it is a Banachspace, and it is reflexive and separable for 1 < p < ∞ .While the Sobolev spaces W k,p ( U ) are generally defined on open domains, we of-ten consider the closure U as the domain for spaces of differentiable functions C k ( U )and C ∞ ( U ). For instance, C k ( U ) is the Banach space of k -times differentiable func-tions on U whose derivatives up to order k are bounded and uniformly continuouson U ; note that uniform continuity implies the existence of continuous extensionsto the closure U . Given suitable regularity assumptions for the boundary of U , onecan show (with some effort) that C k ( U ) is precisely the set of functions which admit k -times differentiable extensions to some open set containing U .The following two results are special cases of the more general Theorems A.6and A.9 in Appendix A, proofs of which may be found e.g. in [ AF03 ]. Proposition . Assume ≤ p < ∞ , kp > n and d ≥ is an integer. Then there exists a continuous inclusion W k + d,p ( U ) ֒ → C d ( U ) , which is compact if U is bounded. (cid:3) Proposition . If ≤ p < ∞ and U is bounded, then the natural inclusion W k +1 ,p ( U ) ֒ → W k,p ( U ) is compact. (cid:3) Exercise . Show that Proposition 2.5 fails in general for unbounded do-mains, e.g. for R .The next three results for the case kp > n are proved in § A.2 as corollaries ofthe Sobolev embedding theorem.
Proposition . Suppose ≤ p < ∞ , kp > n and ≤ m ≤ k . Then the product pairing ( f, g ) f g defines a continuous bilinear map W k,p ( U ) × W m,p ( U ) → W m,p ( U ) . In particular, W k,p ( U ) is a Banach algebra. (cid:3) The continuity statements above translate into inequalities between the normsin the respective spaces. For example, continuous inclusions W k + d,p ֒ → C d and W k +1 ,p ֒ → W k,p respectively imply that k f k C d ≤ c k f k W k + d,p k f k W k,p ≤ c k f k W k +1 ,p for some constants c > d , k , p or U , but not f . Similarly,the Banach algebra property implies k f g k W m,p ≤ c k f k W k,p k g k W m,p , where again, the constant c is independent of g and f . Chris Wendl
We state the next result only for the case of bounded domains; it does havean extension to unbounded domains, but the statement becomes more complicated(cf. Theorem A.12). Given an open set Ω ⊂ R n , we denote W k,p ( U , Ω) := n u ∈ W k,p ( U , R n ) (cid:12)(cid:12)(cid:12) u ( U ) ⊂ Ω o . Note that this is an open subset if kp > n , due to the Sobolev embedding theorem.
Proposition C k -continuity property) . Assume ≤ p < ∞ , kp > n , U isbounded and Ω ⊂ R n is an open set. Then the map C k (Ω , R N ) × W k,p ( U , Ω) → W k,p ( U , R N ) : ( f, u ) f ◦ u is well defined and continuous. (cid:3) Remark . Though we will not yet use it in this lecture, Propositions 2.4,2.7 and 2.8 are the essential conditions needed in order to define smooth Banachmanifold structures on spaces of W k,p -smooth maps from one manifold to another,cf. [ El˘ı67 , Pal68 ]. This only works under the condition kp > n , as the smoothcategory is not well equipped to deal with discontinuous maps!The following rescaling result will be needed for nonlinear regularity arguments;see Theorem A.15 in Appendix A for a proof.
Proposition . Assume p ∈ [1 , ∞ ) and k ∈ N satisfy kp > n , let ˚ D n denote the open unit ball in R n , and for each f ∈ W k,p (˚ D n ) and ǫ ∈ (0 , , define f ǫ ∈ W k,p (˚ D n ) by f ǫ ( x ) := f ( ǫx ) . Then there exist constants
C > and r > such that for every f ∈ W k,p (˚ D n ) , k f ǫ − f (0) k W k,p (˚ D n ) ≤ Cǫ r k f − f (0) k W k,p (˚ D n ) for all ǫ ∈ (0 , . (cid:3) Exercise . Working on a 2-dimensional domain with kp >
2, prove directlythat for any multiindex α of positive degree k , k ∂ α f ǫ k L p (˚ D ) ≤ ǫ k − /p k ∂ α f k L p (˚ D ) for f ∈ W k,p (˚ D ). Find examples (e.g. in W , (˚ D )) to show that no estimate of theform k ∂ α f ǫ k L p (˚ D ) ≤ C ǫ k f − f (0) k W k,p (˚ D ) with lim ǫ → + C ǫ = 0 is possible when kp ≤ We will make considerable use of the fact that the linear first-order differentialoperator ¯ ∂ := ∂ s + i∂ t : C ∞ ( C , C ) → C ∞ ( C , C )is elliptic . There is no need to discuss here precisely what ellipticity means in fullgenerality (see [ Wend , § ectures on Symplectic Field Theory Theorem . If < p < ∞ , then ¯ ∂ : W ,p (˚ D ) → L p (˚ D ) admits a boundedright inverse T : L p (˚ D ) → W ,p (˚ D ) . Theorem . If < p < ∞ and k ∈ N , then there exists a constant c > such that for all f ∈ W k,p (˚ D ) , k f k W k,p ≤ c k ¯ ∂f k W k − ,p . Here W k,p (˚ D ) denotes the W k,p -closure of C ∞ (˚ D ), the latter being space ofsmooth functions on ˚ D with compact support.The complete proofs of the two theorems above are rather lengthy, and we shallrefer to [ Wend , § p = 2. First, it is straightforward to show that the function K ∈ L ( C ) defined by K ( z ) = 12 πz is a fundamental solution for the equation ¯ ∂u = f , meaning it satisfies¯ ∂K = δ in the sense of distributions, where δ denotes the Dirac δ -function. Hence for any f ∈ C ∞ ( C ), one finds a smooth solution u : C → C to the equation ¯ ∂u = f as theconvolution u ( z ) = ( K ∗ f )( z ) := Z C K ( z − ζ ) f ( ζ ) dµ ( ζ ) , where dµ ( ζ ) denotes the Lebesgue measure with respect to the variable ζ ∈ C . Itis not hard to show from this formula that whenever f ∈ C ∞ , K ∗ f has decayingbehavior at infinity (see [ Wend , Lemma 2.6.13]). Thus if u ∈ C ∞ and ¯ ∂u = f , itfollows that u − K ∗ f is a holomorphic function on C that decays at infinity, hence u ≡ K ∗ f . Since C ∞ (˚ D ) is dense in L p (˚ D ) for all p < ∞ , Theorem 2.12 now followsfrom the claim that for all f ∈ C ∞ (˚ D ), there exist estimates of the form(2.4) k K ∗ f k L p (˚ D ) ≤ c k f k L p (˚ D ) , k ∂ j ( K ∗ f ) k L p (˚ D ) ≤ c k f k L p (˚ D ) , with ∂ j = ∂ s or ∂ t for j = 1 , c > f . Exercise . Use Theorem 2.12 and the remarks above to prove Theorem 2.13for the case k = 1 with f ∈ C ∞ (˚ D ), then extend it to f ∈ W ,p (˚ D ) by a densityargument. Then extend it to the general case by differentiating both f and ¯ ∂f .The first estimate in (2.4) is not too hard if you remember your introductorymeasure theory class: it follows from a general “potential inequality” for convolu-tion operators (see [ Wend , Lemma 2.6.10]), similar to Young’s inequality, the keypoints being that K is locally of class L and ˚ D has finite measure. For the secondinequality, observe that ¯ ∂ ( K ∗ f ) = f , and the rest of the first derivative of K ∗ f isdetermined by ∂ ( K ∗ f ), where ∂ := ∂ s − i∂ t . Chris Wendl
Differentiating K in the sense of distributions provides a formula for ∂ ( K ∗ f ) as aprincipal value integral, namely ∂ ( K ∗ f )( z ) = − π lim ǫ → + Z | ζ − z |≥ ǫ f ( ζ )( z − ζ ) dµ ( ζ ) . This is a so-called singular integral operator : it is similar to our previous con-volution operator, but more difficult to handle because the kernel z is not of class L on C . The proof of the estimate k ∂ ( K ∗ f ) k L p ≤ c k f k L p for all f ∈ C ∞ (˚ D )follows from a rather difficult general estimate on singular integral operators, knownas the Calder´on-Zygmund inequality , cf. [
Wend , § p = 2.As is the case for all elliptic operators with constant coefficients, the L -estimateon the fundamental solution of ¯ ∂ admits an easy proof using Fourier transforms: Proposition . For all f ∈ C ∞ ( C ) , we have k ∂ ( K ∗ f ) k L = k f k L . Proof.
A sufficiently nice function u : C → C is related to its Fourier transformˆ u : C → C by u ( z ) = Z C ˆ u ( ζ ) e πi ( z · ζ ) dµ ( ζ )and thus satisfies the identities c ¯ ∂u ( ζ ) = 2 πiζ ˆ u ( ζ ) , c ∂u ( ζ ) = 2 πiζ ˆ u ( ζ ) . Since u = K ∗ f we have ˆ u = ˆ K ˆ f , and since ¯ ∂K = δ , we have 2 πiζ ˆ K = 1. Hencewe may apply Plancharel’s theorem to deduce k ∂ ( K ∗ f ) k L = k ∂u k L = k c ∂u k L = k πiζ ˆ u k L = k πiζ ˆ K ˆ f k L = (cid:13)(cid:13)(cid:13)(cid:13) ζζ πiζ ˆ K ˆ f (cid:13)(cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13)(cid:13)(cid:13) ζζ ˆ f (cid:13)(cid:13)(cid:13)(cid:13) L = k ˆ f k L = k f k L . (cid:3) We will now use the estimate k u k W k,p ≤ c k ¯ ∂u k W k − ,p from the previous sectionto prove three types of results about solutions to Cauchy-Riemann type equations:(1) All solutions of reasonable Sobolev-type regularity are smooth.(2) Any collection of solutions satisfying uniform bounds in certain Sobolevnorms also locally satisfy uniform C ∞ -bounds.(3) All reasonable Sobolev-type topologies on spaces of solutions are (locally)equivalent to the C ∞ -topology.In the following, D r ⊂ C ectures on Symplectic Field Theory denotes the closed disk of radius r >
0, and ˚ D r denotes its interior. Note that func-tions of class C ∞ ( D r ) are assumed to be smooth up to the boundary (or equivalently,on some open neighborhood of D r in C ), not just on ˚ D r . Recall from Exercise 2.2 that every linear Cauchy-Riemann type operator on a vector bundle of complex rank n locally takes the form¯ ∂ + A , where ¯ ∂ = ∂ s + i∂ t , and A is a smooth function with values in End R ( C n ).Using the Sobolev embedding theorem, the following result implies by inductionthat solutions u ∈ W ,p to the equation ( ¯ ∂ + A ) u = 0 are always smooth. Theorem . Assume < p < ∞ and k, m ∈ N .(1) If u ∈ W k,p (˚ D ) satisfies ¯ ∂u ∈ W m,p (˚ D ) , then u is in W m +1 ,p on everycompact subset of ˚ D .(2) Suppose f ν ∈ W m,p (˚ D ) is a sequence converging in the W m,p -topology to f ∈ W m,p (˚ D ) as ν → ∞ , and u ν ∈ W k,p (˚ D ) is a sequence with ¯ ∂u ν = f ν .(a) If there exist uniform bounds on k u ν k W k,p and k f ν k W m,p over ˚ D as ν → ∞ , then k u ν k W m +1 ,p is also uniformly bounded on every compactsubset of ˚ D .(b) If the sequence u ν is W k,p -convergent on ˚ D to a function u ∈ W k,p (˚ D ) satisfying ¯ ∂u = f , then it is also W m +1 ,p -convergent on every compactsubset of ˚ D . Proof.
We begin by proving statement (2a), assuming that statement (1) isalready known, hence u ν ∈ W m +1 ,p loc (˚ D ) since f ν ∈ W m,p (˚ D ). Assume m = k ,since there is otherwise nothing to prove. Then by induction, it suffices to showthat uniform bounds on k u ν k W k,p (˚ D ) and k f ν k W k,p (˚ D ) imply a uniform bound on k u ν k W k +1 ,p (˚ D r ) for any given r <
1; equivalently, this would mean there is a uniformbound on k ∂ j u ν k W k,p (˚ D r ) for j = 1 ,
2. In order to apply the elliptic estimate, we needto work with functions with compact support in ˚ D , thus choose a smooth bumpfunction β ∈ C ∞ (˚ D , [0 , β | D r ≡
1. We then have β ∂ j u ν ∈ C ∞ (˚ D ), so by Theorem 2.13, k ∂ j u ν k W k,p (˚ D r ) ≤ k β ∂ j u ν k W k,p (˚ D ) ≤ c (cid:13)(cid:13) ¯ ∂ ( β ∂ j u ν ) (cid:13)(cid:13) W k − ,p (˚ D ) ≤ c k ( ¯ ∂β )( ∂ j u ν ) k W k − ,p + c k β ¯ ∂ ( ∂ j u ν ) k W k − ,p . (2.5)The first term on the right hand side is uniformly bounded since ¯ ∂β is smooth and k u ν k W k,p is uniformly bounded. To control the second term, we differentiate theequation ¯ ∂u ν = f ν , giving ¯ ∂ ( ∂ j u ν ) = ∂ j f ν . This also has a uniformly bounded W k − ,p -norm since k f ν k W k,p is uniformly bounded.Since β is smooth, this bounds the second term on the right hand side of (2.5) as ν → ∞ , and we are done.Statement (2b) follows by a similar argument bounding k ∂ j ( u − u ν ) k W k,p (˚ D r ) interms of k u − u ν k W k,p (˚ D ) and k f − f ν k W k,p (˚ D ) ; we leave the details as an exercise. Chris Wendl
Lastly, we prove statement (1), where again it suffices to assume ¯ ∂u = f ∈ W k,p (˚ D ) and show that u | ˚ D r ∈ W k +1 ,p (˚ D r ) for some r <
1. The idea is to use thesame argument that was used for statement (2a), but with the partial derivatives ∂ j u replaced by the difference quotients D hj u ( z ) := u ( z + he j ) − u ( z ) h , j = 1 , , where e := ∂ s , e := ∂ t , and the role of the index ν → ∞ is now played by theparameter h ∈ R \ { } approaching 0. Note that if u ∈ W k,p (˚ D ), then β D hj u is awell-defined function on ˚ D for all | h | 6 = 0 sufficiently small and belongs to W k,p (˚ D ).The analogue of (2.5) in this context is then k D hj u k W k,p (˚ D r ) ≤ k βD hj u k W k,p (˚ D ) ≤ c (cid:13)(cid:13) ¯ ∂ (cid:0) β D hj u (cid:1)(cid:13)(cid:13) W k − ,p (˚ D ) ≤ c k ( ¯ ∂β )( D hj u ) k W k − ,p + c k β ¯ ∂ ( D hj u ) k W k − ,p . The first term is bounded independently of h since ∂ j u ∈ W k − ,p (˚ D ), implying auniform W k − ,p -bound on D hj u as h →
0. To control the second term, we can applythe operator D hj to the equation ¯ ∂u = f , giving¯ ∂ ( D hj u ) = D hj ( ¯ ∂u ) = D hj f. This satisfies a W k − ,p -bound that is uniform in h since ∂ j f ∈ W k − ,p (˚ D ), so weconclude that for all | h | sufficiently small, k D hj u k W k,p (˚ D r ) ≤ c for some constant c > h →
0. By a standard applicationof the Banach-Alaoglu theorem (cf. [
Eva98 , § h ν → D h ν j u is W k,p -convergent on ˚ D r , and its limit is necessarily ∂ j u , which therefore belongs to W k,p . Indeed, if k = 0, the uniform L p -boundon D h ν j u over ˚ D r for any sequence h ν → L p -convergentsubsequence via the Banach-Alaoglu theorem. The limit of this subsequence belongsto L p (˚ D r ), and it is straightforward to show using the definition of weak derivativesthat this limit is ∂ j u . One finds the same result for any k ∈ N by applying thisargument to higher-order derivatives of ∂ j u . The conclusion is that u is in W k +1 ,p on ˚ D r , since u and both of its first partial derivatives belong to W k,p . (cid:3) Exercise . Show that all three parts of Theorem 2.16 continue to hold ifthe operator ¯ ∂ is replaced by ¯ ∂ + A or ¯ ∂ + A ν , where A, A ν ∈ C ∞ ( D , End R ( C n ))with A ν → A in C ∞ as ν → ∞ . Exercise . Use Theorem 2.16(1) to extend Theorem 2.12 to the existenceof a bounded right inverse for¯ ∂ : W k,p (˚ D ) → W k − ,p (˚ D ) . Hint: For any
R > , there exists a bounded linear extension operator E : W k,p (˚ D ) → W k,p (˚ D R ) with the property ( Ef ) | ˚ D = f for all f ∈ W k,p (˚ D ) ; see Theorem A.4 andCorollary A.5. ectures on Symplectic Field Theory The above exercise can be used to improve the first part of Theorem 2.16 tocover weak solutions of class L . We start with a classical result about “weaklyholomorphic” functions: Lemma . If u ∈ L (˚ D ) satisfies ¯ ∂u = 0 in the sense of distributions, then u is smooth and holomorphic. Proof.
Taking real and imaginary parts, it suffices to prove that the samestatement holds for the Laplace equation. By mollification, any weakly harmonicfunction can be approximated in L with smooth harmonic functions. The lat-ter satisfy the mean value property, which behaves well under L -convergence, sothe result follows from the mean value characterization of harmonic functions; see[ Wend , Lemma 2.6.26] for more details. (cid:3)
Lemma . Suppose < p < ∞ , k ∈ N , and u ∈ L (˚ D ) is a weak solution to ¯ ∂u = f for some f ∈ W k,p (˚ D ) . Then u is of class W k +1 ,p on every compact subsetof ˚ D . Proof.
Let T : W k,p (˚ D ) → W k +1 ,p (˚ D ) denote a bounded right inverse of ¯ ∂ : W k +1 ,p (˚ D ) → W k,p (˚ D ) as provided by Exercise 2.18. Then u − T f ∈ L (˚ D ) is aweak solution to ¯ ∂ ( u − T f ) = 0 and is thus smooth by Lemma 2.19. In particular, u − T f restricts to ˚ D r for every r < W k +1 ,p , implying that u also has a restriction in W k +1 ,p (˚ D r ). (cid:3) Corollary . Suppose < p < ∞ . Then given A ∈ C ∞ ( D , End R ( C n )) , every weak solution u ∈ L p (˚ D , C n ) of ( ¯ ∂ + A ) u = 0 is smoothon ˚ D . (cid:3) Locally, every J -holomorphic curve can be re-garded as a map u : ˚ D → C n satisfying u (0) = 0 and¯ ∂ J u := ∂ s u + J ( u ) ∂ t u = 0 , where J is a smooth almost complex structure on C n satisfying J (0) = i . Theo-rem 2.16 now has the following analogue. Theorem . Assume < p < ∞ and k ∈ N satisfy kp > , and fix a smooth almost complex structure J on C n with J (0) = i .(1) Every map u ∈ W k,p (˚ D , C n ) satisfying u (0) = 0 and ¯ ∂ J u = 0 is smoothon ˚ D .(2) Suppose J ν is a sequence of smooth almost complex structures on C n con-verging in C ∞ loc to J as ν → ∞ , and u ν ∈ W k,p (˚ D , C n ) is a sequence ofsmooth maps satisfying ¯ ∂ J ν u ν = 0 .(a) If the maps u ν are uniformly W k,p -bounded on ˚ D , then they are alsouniformly C m -bounded on compact subsets of ˚ D for every m ∈ N .(b) If the sequence u ν is W k,p -convergent on ˚ D to a smooth map u : ˚ D → C n , then it is also C ∞ -convergent on every compact subset of ˚ D . Chris Wendl
Our proof of this will follow much the same outline as the proof of Theorem 2.16,and indeed, one could use exactly the same argument if J were identically equal to i (in which case the theorem can also be deduced from complex analysis). The reasonit works in the general case is that if we zoom in on a sufficiently small neighborhoodof the origin in C n , then J can be viewed as a C ∞ -small perturbation of i . To makethis precise, we shall use the following rescaling trick.Associate to any smooth almost complex structure J on C n the function Q := i − J ∈ C ∞ ( C n , End R ( C n )) . In terms of Q , the equation ∂ s u + J ( u ) ∂ t u = 0 then becomes(2.6) ¯ ∂u − ( Q ◦ u ) ∂ t u = 0 , where we are regarding Q ◦ u as a function ˚ D → End R ( C n ). Given constants R ≥ ǫ ∈ (0 , J and u the functions b J : C n → End R ( C n ) , b J ( p ) := J ( p/R ) , b Q : C n → End R ( C n ) , b Q ( p ) := Q ( p/R ) = i − b J ( p ) , ˆ u : ˚ D → C n , u ( z ) := Ru ( ǫz ) . (2.7)Now u satisfies (2.6) if and only if ˆ u satisfies(2.8) ¯ ∂ ˆ u − ( b Q ◦ ˆ u ) ∂ t ˆ u = 0 . The rescaled almost complex structure has the convenient feature that if J (0) = i ,then b J can be made arbitrarily C ∞ -close to i on the unit disk D n ⊂ C n by choosing R sufficiently large, which means k b Q k C m ( D n ) can be made arbitrarilysmall for every m ∈ N . If u is also continuous and satisfies u (0) = 0, then afterfixing some large value for R , we can also choose ǫ ∈ (0 ,
1] sufficiently small to ensure u (˚ D ) ⊂ ˚ D n and make k b Q ◦ ˆ u k C ( D ) arbitrarily small. By Propositions 2.8 and 2.10,we can similarly arrange for k b Q ◦ ˆ u k W k,p to be arbitrarily small if u is of class W k,p with kp >
2, and the same will hold for k b Q ν ◦ ˆ u ν k W k,p when ν is large if k u ν k W k,p is uniformly bounded and u ν (0) →
0. Here of course we abbreviate Q ν := i − J ν and b Q ν ( p ) := Q ν ( p/R ). The effect is to make equations such as (2.8) W k,p -closeto the linear equation ¯ ∂ ˆ u = 0 if ǫ > R > k ˆ u ν k W k,p (˚ D ) for some sequence u ν , then the resulting W k +1 ,p -bound for u ν will be valid only on ˚ D ǫ , a very small ball about the origin. But thisis good enough for obtaining estimates over all compact subsets of ˚ D : indeed, wecan always reparametrize u : ˚ D → C n to put the origin at some other point andprove suitable estimates near that point, appealing in the end to the fact that anycompact subset of ˚ D is covered by a finite union of small disks about points.The need to use this rescaling trick is one of a few reasons why the condition kp > ectures on Symplectic Field Theory Proof of Theorem 2.22.
We will prove statement (2a) and leave the rest asexercises.By the remarks above, it suffices to prove that if u ν : ˚ D → C n are smooth J ν -holomorphic curves satisfying a uniform bound in W k,p (˚ D ), then for some r <
1, therescaled b J ν -holomorphic curves ˆ u ν : ˚ D → C n defined as in (2.7) satisfy a uniform W k +1 ,p -bound on ˚ D r . In fact, it suffices to prove that every subsequence of u ν hasa further subsequence for which this is true. Indeed, if the bound for the wholesequence did not exist, then we would be able to find a subsequence with normsblowing up to infinity, and no further subsequence of this subsequence could satisfya uniform bound. With this understood, we can appeal to the fact that W k,p -bounded sequences are also C -bounded for kp > u ν with asubsequence (still denoted by u ν ) such that, after a suitable change of coordinateson C n , u ν (0) → . Our goal is then to show that for a suitable choice of the rescaling parameters ǫ and R , this subsequence admits a uniform bound on k ∂ j ˆ u ν k W k,p (˚ D r ) for j = 1 , β ∈ C ∞ (˚ D , [0 , β | D r ≡
1. We then have β ∂ j ˆ u ν ∈ C ∞ (˚ D ), so by Theorem 2.13,(2.9) k ∂ j ˆ u ν k W k,p (˚ D r ) ≤ k β ∂ j ˆ u ν k W k,p (˚ D ) ≤ c (cid:13)(cid:13) ¯ ∂ ( β ∂ j ˆ u ν ) (cid:13)(cid:13) W k − ,p (˚ D ) . Instead of rewriting ¯ ∂ ( β ∂ j ˆ u ν ) as a sum of two terms, let us derive a PDE satisfiedby β ∂ j ˆ u ν . Differentiating the equation ¯ ∂ ˆ u ν − ( b Q ν ◦ ˆ u ν ) ∂ t ˆ u ν = 0 gives¯ ∂ ( ∂ j ˆ u ν ) = ∂ j ( ¯ ∂ ˆ u ν ) = ( d b Q ν ◦ ˆ u ν ) ( ∂ j ˆ u ν , ∂ t ˆ u ν ) + ( b Q ν ◦ ˆ u ν ) ∂ j ∂ t ˆ u ν , thus β ∂ j ˆ u ν satisfies¯ ∂ ( β ∂ j ˆ u ν ) − ( b Q ν ◦ ˆ u ν ) ∂ t ( β ∂ j ˆ u ν )= β ( d b Q ν ◦ ˆ u ν )( ∂ j ˆ u ν , ∂ t ˆ u ν ) + (cid:16) ¯ ∂β − ( b Q ν ◦ ˆ u ν ) ∂ t β (cid:17) ∂ j ˆ u ν = ( d b Q ν ◦ ˆ u ν )( β ∂ j ˆ u ν , ∂ t ˆ u ν ) + (cid:16) ¯ ∂β − ( b Q ν ◦ ˆ u ν ) ∂ t β (cid:17) ∂ j ˆ u ν , (2.10)and combining this with (2.9) gives(2.11) k β ∂ j ˆ u ν k W k,p ≤ c (cid:13)(cid:13) ( b Q ν ◦ ˆ u ν ) ∂ t ( β ∂ j ˆ u ν ) (cid:13)(cid:13) W k − ,p + c (cid:13)(cid:13) ( d b Q ν ◦ ˆ u ν )( β ∂ j ˆ u ν , ∂ t ˆ u ν ) (cid:13)(cid:13) W k − ,p + c (cid:13)(cid:13)(cid:13)(cid:16) ¯ ∂β − ( b Q ν ◦ ˆ u ν ) ∂ t β (cid:17) ∂ j ˆ u ν (cid:13)(cid:13)(cid:13) W k − ,p . In order to find bounds for the three terms on the right, recall that using Proposi-tions 2.8 and 2.10 and the assumption u ν (0) →
0, we can suppose (cid:13)(cid:13)(cid:13) b Q ν ◦ ˆ u ν (cid:13)(cid:13)(cid:13) W k,p ≤ δ Chris Wendl for sufficiently large ν , where δ > ǫ and R . This provides auniform bound on the third term in (2.11), as there is also a continuous productpairing W k,p × W k − ,p → W k − ,p by Prop. 2.7, giving an estimate of the form (cid:13)(cid:13)(cid:13)(cid:16) ¯ ∂β − ( b Q ν ◦ ˆ u ν ) ∂ t β (cid:17) ∂ j ˆ u ν (cid:13)(cid:13)(cid:13) W k − ,p ≤ c (cid:13)(cid:13)(cid:13)(cid:16) ¯ ∂β − ( b Q ν ◦ ˆ u ν ) ∂ t β (cid:17)(cid:13)(cid:13)(cid:13) W k,p · k ∂ j ˆ u ν k W k − ,p ≤ c ′ k ˆ u ν k W k,p ≤ c ′′ . For the first term on the right side of (2.11), the product pairing similarly gives (cid:13)(cid:13) ( b Q ν ◦ ˆ u ν ) ∂ t ( β ∂ j ˆ u ν ) (cid:13)(cid:13) W k − ,p ≤ c (cid:13)(cid:13) b Q ν ◦ ˆ u ν (cid:13)(cid:13) W k,p · k ∂ t ( β ∂ j ˆ u ν ) k W k − ,p ≤ cδ k β ∂ j ˆ u ν k W k,p . Finally, since J ν → J in C k +1 on compact subsets, we are also free to assume afteradjusting the rescaling parameters that k d b Q ν ◦ ˆ u ν k W k,p ≤ δ, so we can apply the product pairing W k,p × W k − ,p → W k − ,p twice to estimate (cid:13)(cid:13) ( d b Q ν ◦ ˆ u ν )( β ∂ j ˆ u ν , ∂ t ˆ u ν ) (cid:13)(cid:13) W k − ,p ≤ c (cid:13)(cid:13) d b Q ν ◦ ˆ u ν (cid:13)(cid:13) W k,p · k β ∂ j ˆ u ν k W k,p · k ∂ t ˆ u ν k W k − ,p ≤ cδ k β ∂ j ˆ u ν k W k,p · k ˆ u ν k W k,p ≤ cc ′ δ k β ∂ j ˆ u ν k W k,p =: c ′′ δ k β ∂ j ˆ u ν k W k,p . Combining the three estimates for the right hand side of (2.11) now gives k β ∂ j ˆ u ν k W k,p ≤ c + cδ k β ∂ j ˆ u ν k W k,p , so after adjusting the scaling parameters R and ǫ to ensure cδ <
1, we obtain theuniform bound k β ∂ j ˆ u ν k W k,p ≤ c − cδ . This provides the desired uniform bound on k ∂ j ˆ u ν k W k,p (˚ D r ) . (cid:3) Exercise . Use an analogous argument via difference quotients to provestatement (1) in Theorem 2.22.
Hint: If you’re anything like me, you might getstuck trying to estimate the second term in the difference quotient analogue of (2.11) . The difficulty is that this expression was derived using the chain rule forderivatives, and there is no similarly simple chain rule for difference quotients. Thetrick is to remember that difference quotients only differ from the correspondingderivatives by a remainder term. The remainder will produce an extra term in thedifference quotient version of (2.11) , but the extra term can be bounded.
The following lemma can be applied in the case A ∈ C ∞ ( D , End C ( C n )) to provethe aforementioned standard fact that complex-linear Cauchy-Riemann type oper-ators induce holomorphic structures on vector bundles. The version with weakenedregularity will be applied below to prove a useful “unique continuation” result aboutsolutions to ( ¯ ∂ + A ) f = 0 in the real-linear case. ectures on Symplectic Field Theory Lemma . Assume < p < ∞ and A ∈ L p (˚ D , End R ( C n )) . Then for suffi-ciently small ǫ > , the problem ¯ ∂u + Au = 0 u (0) = u has a solution u ∈ W ,p (˚ D ǫ , C n ) . Remark . Note that u : ˚ D ǫ → C n in the above statement is only a weak solution to ¯ ∂u + Au = 0, as it is not necessarily differentiable, but by the Sobolevembedding theorem, it is at least continuous. Proof of Lemma 2.24.
The main idea is that if we take ǫ > ∂ + A to ˚ D ǫ can be regarded as a small perturbationof ¯ ∂ in the space of bounded linear operators W ,p → L p . Since the latter has abounded right inverse by Theorem 2.12, the same will be true for the perturbation.Since p >
2, the Sobolev embedding theorem implies that functions u ∈ W ,p are also continuous and bounded by k u k W ,p , thus we can define a bounded linearoperator Φ : W ,p (˚ D ) → L p (˚ D ) × C n : u ( ¯ ∂u, u (0)) . Theorem 2.12 implies that this operator is also surjective and has a bounded rightinverse, namely L p (˚ D ) × C n → W ,p (˚ D ) : ( f, u ) T f − T f (0) + u , where T : L p (˚ D ) → W ,p (˚ D ) is a right inverse of ¯ ∂ . Thus any operator sufficientlyclose to Φ in the norm topology also has a right inverse. Now define χ ǫ : D → R tobe the function that equals 1 on D ǫ and 0 outside of it, and letΦ ǫ : W ,p (˚ D ) → L p (˚ D ) × C n : u (( ¯ ∂ + χ ǫ A ) u, u (0)) . To see that this is a bounded operator, it suffices to check that W ,p → L p : u Au is bounded if A ∈ L p ; indeed, k Au k L p ≤ k A k L p k u k C ≤ c k A k L p k u k W ,p , again using the Sobolev embedding theorem. Now by this same trick, we find k Φ ǫ u − Φ u k = k χ ǫ Au k L p (˚ D ) ≤ c k A k L p (˚ D ǫ ) k u k W ,p (˚ D ) , thus k Φ ǫ − Φ k is small if ǫ is small, and it follows that in this case Φ ǫ is surjective.Our desired solution is therefore the restriction of any u ∈ Φ − ǫ (0 , u ) to ˚ D ǫ . (cid:3) Here is a corollary, which says that every solution to a real-linear Cauchy-Riemann type equation looks locally like a holomorphic function in some continuous local trivialization.
Theorem . Suppose A : D → End R ( C n ) is smooth and u : ˚ D → C n satisfies the equation ¯ ∂u + Au = 0 with u (0) = 0 . Then for sufficientlysmall ǫ > , there exist maps Φ ∈ C ( D ǫ , End C ( C n )) and f ∈ C ∞ (˚ D ǫ , C n ) such that u ( z ) = Φ( z ) f ( z ) , ¯ ∂f = 0 , and Φ(0) = . Chris Wendl
Proof.
After shrinking the domain if necessary, we may assume without lossof generality that the smooth solution u : ˚ D → C n is bounded. Choose a map C : D → End C ( C n ) satisfying C ( z ) u ( z ) = A ( z ) u ( z ) and | C ( z ) | ≤ | A ( z ) | for almostevery z ∈ D . Then C ∈ L ∞ (˚ D , End C ( C n )) and u is a weak solution to ( ¯ ∂ + C ) u = 0.Note that since we do not know anything about the zero set of u , we cannot assume C is continuous, but we have no trouble assuming C ∈ L p (˚ D ) for every p > ∂ + C is now complex linear, we can use Lemma 2.24 to find a complex basisof W ,p -smooth weak solutions to ( ¯ ∂ + C ) v = 0 on ˚ D ǫ that define the standard basis of C n at 0, and these solutions are continuous by the Sobolev embedding theorem. Thisgives rise to a map Φ ∈ C (˚ D ǫ , End C ( C n )) that satisfies ( ¯ ∂ + C )Φ = 0 in the senseof distributions and Φ(0) = . Since Φ is continuous, we can assume without loss ofgenerality that Φ( z ) is invertible everywhere on ˚ D ǫ . Setting f := Φ − u : ˚ D ǫ → C n ,the Leibniz rule then implies0 = ( ¯ ∂ + C ) u = ( ¯ ∂ + C )(Φ f ) = (cid:2) ( ¯ ∂ + C )Φ (cid:3) f + Φ( ¯ ∂f ) = Φ( ¯ ∂f ) , thus ¯ ∂f = 0, and f is smooth by Lemma 2.19. (cid:3) Corollary . Suppose D is a linear Cauchy-Riemanntype operator on a vector bundle E over a connected Riemann surface, and η ∈ Γ( E ) satisfies D η = 0 . Then either η is identically zero or its zeroes are isolated. The similarity principle also has many nice applications for the nonlinear Cauchy-Riemann equation. Here is another “unique continuation” type result for the non-linear case.
Proposition . Suppose J is a smooth almost complex structure on C n and u, v : ˚ D → C n are smooth J -holomorphic curves such that u (0) = v (0) = 0 and u and v have matching partial derivatives of all orders at . Then u ≡ v on aneighborhood of . Proof.
Let h = v − u : ˚ D → C n . We have(2.12) ∂ s u + J ( u ( z )) ∂ t u = 0and ∂ s v + J ( u ( z )) ∂ t v = ∂ s v + J ( v ( z )) ∂ t v + [ J ( u ( z )) − J ( v ( z ))] ∂ t v = − [ J ( u ( z ) + h ( z )) − J ( u ( z ))] ∂ t v = − (cid:18)Z ddt J ( u ( z ) + th ( z )) dt (cid:19) ∂ t v = − (cid:18)Z dJ ( u ( z ) + th ( z )) · h ( z ) dt (cid:19) ∂ t v =: − A ( z ) h ( z ) , (2.13)where the last step defines a smooth family of linear maps A ( z ) ∈ End R ( C n ). Sub-tracting (2.12) from (2.13) gives the linear equation ∂ s h ( z ) + ¯ J ( z ) ∂ t h ( z ) + A ( z ) h ( z ) = 0 , where ¯ J ( z ) := J ( u ( z )). This is a linear Cauchy-Riemann type equation on a trivialcomplex vector bundle over ˚ D with complex structure ¯ J ( z ) on the fiber at z . The ectures on Symplectic Field Theory similarity principle thus implies h ( z ) = Φ( z ) f ( z ) near 0 for some holomorphic func-tion f ( z ) ∈ C n and some continuous map Φ( z ) ∈ GL(2 n, R ) representing a change oftrivialization. Now if h has vanishing derivatives of all orders at 0, Taylor’s formulaimplies lim z → | Φ( z ) f ( z ) || z | k = 0for all k ∈ N , so f must also have a zero of infinite order and thus f ≡ (cid:3) We now prove a global result about the structure of closed J -holomorphic curves.In Lecture 6 we will be able to generalize it in a straightforward way for puncturedholomorphic curves with asymptotically cylindrical behavior. Theorem . Assume (Σ , j ) is a closed connected Riemann surface, ( W, J ) is a smooth almost complex manifold and u : (Σ , j ) → ( W, J ) is a nonconstantpseudoholomorphic curve. Then there exists a factorization u = v ◦ ϕ , where • ϕ : (Σ , j ) → (Σ ′ , j ′ ) is a holomorphic map of positive degree to anotherclosed and connected Riemann surface (Σ ′ , j ′ ) ; • v : (Σ ′ , j ′ ) → ( W, J ) is a pseudoholomorphic curve which is embedded exceptat a finite set of critical points and self-intersections. Note that holomorphic maps (Σ , j ) → (Σ ′ , j ′ ) of degree 1 are always diffeomor-phisms, so the factorization u = v ◦ ϕ in this case is just a reparametrization, and u is then called a simple curve. In all other cases, k := deg( ϕ ) ≥ ϕ is ingeneral a branched cover; we then call u a k -fold branched cover of the simplecurve v .The main idea in the proof is to construct Σ ′ (minus some punctures) explicitlyas the image of u after removing finitely many singular points, so that we can take v to be the inclusion Σ ′ ֒ → W . The map ϕ : Σ → Σ ′ is then uniquely determined.In order to carry out this program, we need some information on what the imageof u can look like near each of its singularities. These come in two types, each typecorresponding to one of the lemmas below, both of which should seem immediatelyplausible if your intuition comes from complex analysis. Lemma . Suppose u : (Σ , j ) → ( W, J ) and v : (Σ ′ , j ′ ) → ( W, J ) are two nonconstant pseudoholomorphic curves with an intersection u ( z ) = v ( z ′ ) . Then there exist neighborhoods z ∈ U ⊂ Σ and z ′ ∈ U ′ ⊂ Σ ′ such thateither u ( U ) = v ( U ′ ) or u ( U \ { z } ) ∩ v ( U ′ ) = u ( U ) ∩ v ( U ′ \ { z ′ } ) = ∅ . (cid:3) Lemma . Suppose u : (Σ , j ) → ( W, J ) is a nonconstant pseudo-holomorphic curve and z ∈ Σ is a critical point of u . Then a neighborhood U ⊂ Σ of z can be biholomorphically identified with the unit disk D ⊂ C such that u ( z ) = v ( z k ) for z ∈ D = U , where k ∈ N , and v : D → W is an injective J -holomorphic map with no criticalpoints except possibly at the origin. (cid:3) Chris Wendl
These two local results follow from a well-known formula of Micallef and White[
MW95 ] describing the local behavior of J -holomorphic curves near critical pointsand their intersections. The proof of that theorem is analytically quite involved, butone can also use an easier “approximate” version, which is proved in [ Wend , § J is always a small perturbationof i , hence the local behavior of J -holomorphic curves is also similar to the integrablecase. Proof of Theorem 2.29.
Let Crit( u ) = { z ∈ Σ | du ( z ) = 0 } denote the setof critical points, and define ∆ ⊂ Σ to be the set of all points z ∈ Σ such that thereexists z ′ ∈ Σ and neighborhoods z ∈ U ⊂ Σ and z ′ ∈ U ′ ⊂ Σ with u ( z ) = u ( z ′ ) but u ( U \ { z } ) ∩ u ( U ′ \ { z ′ } ) = ∅ .The lemmas quoted above imply that both of these sets are discrete. Both aretherefore finite, and the set ˙Σ ′ = u (Σ \ (Crit( u ) ∪ ∆)) ⊂ W is then a smoothsubmanifold of W with J -invariant tangent spaces, so it inherits a natural complexstructure j ′ for which the inclusion ( ˙Σ ′ , j ′ ) ֒ → ( W, J ) is pseudoholomorphic. Weshall now construct a new Riemann surface (Σ ′ , j ′ ) from which ( ˙Σ ′ , j ′ ) is obtainedby removing a finite set of points. Let b ∆ = (Crit( u ) ∪ ∆) / ∼ , where two pointsin Crit( u ) ∪ ∆ are defined to be equivalent whenever they have neighborhoods inΣ with identical images under u . Then for each [ z ] ∈ b ∆, the branching lemmaprovides an injective J -holomorphic map u [ z ] from the unit disk D onto the imageof a neighborhood of z under u . We define (Σ ′ , j ′ ) byΣ ′ = ˙Σ ′ ∪ Φ G [ z ] ∈ b ∆ D , where the gluing map Φ is the disjoint union of the maps u [ z ] : D \ { } → ˙Σ ′ for each[ z ] ∈ b ∆; since this map is holomorphic, the complex structure j ′ extends from ˙Σ ′ toΣ ′ . Combining the maps u [ z ] : D → W with the inclusion ˙Σ ′ ֒ → W now defines apseudoholomorphic map v : (Σ ′ , j ′ ) → ( W, J ) which restricts to ˙Σ ′ as an embeddingand otherwise has at most finitely many critical points and double points. Moreover,the restriction of u to Σ \ (Crit( u ) ∪ ∆) defines a holomorphic map to ( ˙Σ ′ , j ′ ) whichextends by removal of singularities to a proper holomorphic map ϕ : (Σ , j ) → (Σ ′ , j ′ )such that u = v ◦ ϕ . Its holomorphicity implies that it has positive degree. (cid:3) ECTURE 3
Asymptotic operatorsContents
We now begin with the analysis of the particular class of J -holomorphic curvesthat are important in SFT. The next three lectures will focus on the linearizedproblem, the goal being to prove that this linearization is Fredholm and to computeits index. Using this along with the implicit function theorem and the Sard-Smaletheorem (on genericity of smooth nonlinear Fredholm maps), we will later be ableto show that moduli spaces of asymptotically cylindrical J -holomorphic curves aresmooth finite-dimensional manifolds under suitable genericity assumptions. Since Morse homology is the prototype for all Floer-type theories, we can gainuseful intuition by recalling how the analysis works for the linearization of the gradi-ent flow problem in Morse theory. The basic features of the problem were discussedalready in § M, g ) is a closed n -dimensional Riemannian manifold, f : M → R is asmooth function, and for two critical points x + , x − ∈ Crit( f ), consider the modulispace of parametrized gradient flow lines M ( x − , x + ) := (cid:26) u ∈ C ∞ ( R , M ) (cid:12)(cid:12) ˙ u + ∇ f ( u ) = 0 , lim s →±∞ u ( s ) = x ± (cid:27) . The map M ( x − , x + ) → M : u u (0) gives a natural identification of M ( x − , x + )with the intersection between the unstable manifold of x − and the stable manifoldof x + for the negative gradient flow, and we say the pair ( g, f ) is Morse-Smale if f is Morse and this intersection is transverse, in which case M ( x − , x + ) is a smoothmanifold with dim M ( x − , x + ) = ind( x − ) − ind( x + ) . Chris Wendl
This can all be proved using finite-dimensional differential topology, but since thatapproach does not work in the study of Floer trajectories or holomorphic curvesin symplectizations, let us instead see how one proves it using nonlinear functionalanalysis. For more details on the following discussion, see [
Sch93 ].Following the strategy laid out in § M ( x − , x + ) can be identified with thezero set of a smooth section σ : B → E : u ˙ u + ∇ f ( u ) , where B is a Banach manifold of maps u : R → M satisfying lim s →±∞ u ( s ) = x ± ,and E → B is a smooth Banach space bundle whose fibers E u contain Γ( u ∗ T M ).The linearization D σ ( u ) : T u B → E u of this section at a zero u ∈ σ − (0) defines afirst-order linear differential operator D u : Γ( u ∗ T M ) → Γ( u ∗ T M )which takes the form D u η = ∇ s η + ∇ η ∇ f for any choice of symmetric connection ∇ on M . Taking suitable Sobolev comple-tions of Γ( u ∗ T M ), we are therefore led to consider bounded linear operators of theform(3.1) D u = ∇ s + ∇∇ f : W k,p ( u ∗ T M ) → W k − ,p ( u ∗ T M )for k ∈ N and 1 < p < ∞ , and the first task is to prove that whenever x + and x − satisfy the Morse condition, this is a Fredholm operator of index ind D u =ind( x − ) − ind( x + ).Choose coordinates near x + in which g looks like the standard Euclidean innerproduct at x + . This induces a trivialization of u ∗ T M over [ T, ∞ ) for T > ∇ is the standardone determined by these coordinates on [ T, ∞ ). Using the trivialization to identifysections β ∈ Γ( u ∗ T M ) over [ T, ∞ ) with maps f : [ T, ∞ ) → R n , D u now acts on f as(3.2) ( D u f )( s ) = ∂ s f ( s ) + A ( s ) f ( s ) , where A ( s ) ∈ R n × n is the matrix of the linear transformation dX ( s ) : R n → R n ,with X ( s ) ∈ R n being the coordinate representation of ∇ f ( u ( s )) ∈ T u ( s ) M . As s → ∞ , the zeroth-order term in this expression converges to a symmetric matrix A + := lim s →∞ A ( s ) , which is the coordinate representation of the Hessian ∇ f ( x + ). Any choice of coor-dinates near x − produces a similar formula for D u over ( −∞ , − T ], A ( s ) convergingas s → −∞ to another symmetric matrix A − representing ∇ f ( x − ). Both theMorse condition and the dimension ind( x − ) − ind( x + ) can now be expressed entirely We are ignoring an analytical subtlety: since u ∗ T M → R has no canonical trivialization and R is noncompact, it is not completely obvious what the definition of the Sobolev space W k,p ( u ∗ T M )should be. We will return to this issue in a more general context in the next lecture. ectures on Symplectic Field Theory in terms of these two matrices: x ± is Morse if and only if A ± is invertible, and theFredholm index of D u will then beind( x − ) − ind( x + ) = dim E − ( A − ) − dim E − ( A + ) , where for any symmetric matrix A we denote by E − ( A ) the direct sum of all itseigenspaces with negative eigenvalue. The main linear functional analytic resultunderlying Morse homology can now be stated as follows (cf. [ Sch93 ]):
Proposition . Assume k ∈ N and < p < ∞ . Suppose E → R is asmooth vector bundle with trivializations fixed in neighborhoods of −∞ and + ∞ , and D : W k,p ( E ) → W k − ,p ( E ) is a first-order differential operator which asymptoticallytakes the form (3.2) near ±∞ with respect to the chosen trivializations, where A ( s ) is a smooth family of n -by- n matrices with well-defined asymptotic limits A ± :=lim s →±∞ A ( s ) which are symmetric. If A + and A − are also invertible, then D isFredholm and (3.3) ind( D ) = dim E − ( A − ) − dim E − ( A + ) . (cid:3) Remark . The hypothesis that A ± is invertible in Prop. 3.1 cannot be lifted:indeed, suppose D is Fredholm but e.g. A + has 0 in its spectrum. Then one caneasily perturb A ( s ) and hence A + in two distinct ways producing two distinct valuesof dim E − ( A + ), pushing the zero eigenvalue either up or down. This produces twoperturbed Fredholm operators that have different indices according to (3.3), butthey also belong to a continuous family of Fredholm operators, and must thereforehave the same index, giving a contradiction.The formula (3.3) makes sense of course because E − ( A ± ) are both finite-dimen-sional vector spaces, but in Floer-type theories we typically encounter critical pointswith infinite Morse index. With this in mind, it is useful to note that (3.3) canbe rewritten without explicitly referencing E − ( A + ) or E − ( A − ). Indeed, choosea continuous path of symmetric matrices { B t } t ∈ [ − , connecting B ( −
1) := A − to B (1) := A + . The spectrum of B t varies continuously with t in the following sense:one can choose a family of continuous functions { λ j : [ − , → R } j ∈ I for the index set I = { , . . . , n } such that for every t ∈ [ − , B t counted with multiplicity is { λ j ( t ) } j ∈ I . The spectral flow from A − to A + isthen defined as a signed count of the number of paths of eigenvalues that cross fromone side of zero to the other, namely (cf. Theorem 3.3) µ spec ( A − , A + ) := (cid:8) j ∈ I (cid:12)(cid:12) λ j ( − < < λ j (1) (cid:9) − (cid:8) j ∈ I (cid:12)(cid:12) λ j ( − > > λ j (1) (cid:9) . The index formula (3.3) now becomesind( D ) = µ spec ( A − , A + ) . This description of the index has the advantage that it could potentially makesense and give a well-defined integer even if A ± were symmetric operators on aninfinite-dimensional Hilbert space: they might both have infinitely many positive Chris Wendl and negative eigenvalues, but only finitely many that change sign along a path from A − to A + . We will make this discussion precise in the next section. We will see in § T x M → T x M defined by the Hessian ∇ f ( x ) of a Morse function f : M → R at a critical point is played by a certain class of symmetric operators onthe space of loops η : S → R n , namely operators of the form(3.4) ( A η )( t ) := − J ∂ t η ( t ) − S ( t ) η ( t ) , where J denotes the standard complex structure on R n = C n , and S : S → End( R n ) is a smooth loop of symmetric matrices. The goal of this section is todefine a notion of spectral flow for operators of this type. Regarding A as anunbounded linear operator on L ( S , R n ) with dense domain H ( S , R n ), we willsee that its spectrum consists of isolated real eigenvalues with finite multiplicity. Weshall prove: Theorem . Assume { S s : S → End( R n ) } s ∈ [ − , is a smooth family of loopsof symmetric matrices, and consider the corresponding -parameter family of un-bounded linear operators A s = − J ∂ t − S s ( t ) : L ( S , R n ) ⊃ H ( S , R n ) → L ( S , R n ) . Then there exists a set of continuous functions { λ j : [ − , → R } j ∈ Z such that for every s ∈ [ − , , the spectrum of A s consists of the numbers { λ j ( s ) } j ∈ Z ,each of which is an eigenvalue with finite multiplicity equal to the number of timesit is repeated as j varies in Z .Moreover, if additionally A − := A − and A + := A both have trivial kernel,then the number µ spec ( A − , A + ) ∈ Z defined by (cid:8) j ∈ Z (cid:12)(cid:12) λ j ( − < < λ j (1) (cid:9) − (cid:8) j ∈ Z (cid:12)(cid:12) λ j ( − > > λ j (1) (cid:9) is well defined and depends only on A − and A + . We will start by giving a more abstract definition of spectral flow as an inter-section number between a path of symmetric index 0 Fredholm operators and thesubvariety of noninvertible operators. This relies on the general fact that spacesof operators with kernel and cokernel of fixed finite dimensions form smooth finite-codimensional submanifolds in the Banach space of all bounded linear operators.We explain this fact in § § § § R , leading to a proof of Theorem 3.3.Spectral flow can be defined more generally for certain classes of self-adjointelliptic partial differential operators, see e.g. [ APS76 , RS95 ], and standard proofs ectures on Symplectic Field Theory of its existence typically rely on perturbation results as in [ Kat95 ] for the spectra ofself-adjoint operators. In the following presentation, we have chosen to avoid makingexplicit use of self-adjointness and instead focus on the Fredholm property; in thisway the discussion is mostly self-contained and, in particular, does not require anyresults from [
Kat95 ]. Fix a field F := R or C . Given Banach spaces X and Y over F , denote by L F ( X, Y ) the Banach space ofbounded F -linear maps from X to Y , with L F ( X ) := L F ( X, X ), and letFred F ( X, Y ) ⊂ L F ( X, Y )denote the open subset consisting of Fredholm operators. Recall that an operator T ∈ L F ( X, Y ) is
Fredholm if its image is closed, and its kernel and cokernel(i.e. the quotient coker T := Y / im T ) are both finite dimensional. Its index isdefined as ind F ( T ) := dim F ker T − dim F coker T ∈ Z . The index defines a continuous and thus locally constant function Fred F ( X, Y ) → Z ,and for each i ∈ Z , we shall denoteFred i F ( X, Y ) := (cid:8) T ∈ Fred F ( X, Y ) (cid:12)(cid:12) ind( T ) = i (cid:9) . We will often have occasion to use the following general construction. Given T ∈ Fred F ( X, Y ), one can choose splittings into closed linear subspaces X = V ⊕ K, Y = W ⊕ C such that K = ker T , W = im T , the quotient projection π C : Y → coker T restricts to C ⊂ Y as an isomorphism, and T | V defines an isomorphism from V to W . Using these splittings, any other T ∈ Fred F ( X, Y ) can be written in blockform as T = (cid:18) A BC D (cid:19) , with T itself written in this way as (cid:18) A
00 0 (cid:19) for some Banach space isomorphism A : V → W . Let O ⊂
Fred F ( X, Y ) denote the open neighborhood of T for whichthe block A is invertible, and define a map(3.5) Φ : O →
Hom F (ker T , coker T ) : T D − CA − B . Lemma . The map Φ in (3.5) is smooth, and holomorphic in the case F = C ,and its derivative at T defines a surjective bounded linear operator L F ( X, Y ) → Hom F (ker T , coker T ) of the form d Φ( T ) H = π C H | ker T , It is not strictly necessary to require that im T ⊂ Y be closed, as this follows from thefinite-dimensionality of the kernel and cokernel, cf. [ AA02 , Cor. 2.17]. Chris Wendl where π C denotes the natural projection Y → coker T . Moreover, there exists asmooth function Ψ :
O → L F ( X ) such that for every T ∈ O , Ψ( T ) : X → X maps ker Φ( T ) ⊂ ker T isomorphically to ker T . Proof.
Smoothness, holomorphicity and the formula for the derivative areeasily verified from the given formula for Φ; in particular, since the blocks B and C both vanish for T = T , we have d Φ( T ) : L F ( X, Y ) → Hom F ( K, C ) (cid:18) A ′ B ′ C ′ D ′ (cid:19) D ′ . The map Ψ :
O → L F ( X ) = L F ( V ⊕ K ) is defined byΨ( T ) = (cid:18) − A − B (cid:19) . For each T , this is an isomorphism; indeed, its inverse is given byΨ( T ) − = (cid:18) A − B (cid:19) . Then T Ψ( T ) = (cid:18) A C Φ( T ) (cid:19) , and since A is invertible, ker T Ψ( T ) = { }⊕ ker Φ( T ). (cid:3) Proposition . For each i ∈ Z and each nonnegative integer k ≥ i , the subset Fred i,k F ( X, Y ) := (cid:8) T ∈ Fred i F ( X, Y ) (cid:12)(cid:12) dim F ker T = k and dim F coker T = k − i (cid:9) admits the structure of a smooth (and complex-analytic if F = C ) finite-codimensionalBanach submanifold of L F ( X, Y ) , with codim F Fred i,k F ( X, Y ) = k ( k − i ) . Proof.
Applying the implicit function theorem to the map Φ from Lemma 3.4endows a neighborhood of T in Φ − (0) ⊂ Fred F ( X, Y ) with the structure of asmooth Banach submanifold withcodim F Φ − (0) = dim F Hom F (ker T , coker T ) = k ( k − i ) . If F = C , then Φ is also holomorphic and Φ − (0) is thus a complex-analytic sub-manifold near T . Now observe that for every T ∈ O ,dim F ker T = dim F ker Φ( T ) ≤ dim F ker T = k, with equality if and only if Φ( T ) = 0, hence, since the index is locally constant, weget Φ − (0) = Fred i,k F ( X, Y ) in a neighborhood of T . (cid:3) Holomorphicity in this infinite-dimensional setting means the same thing as usual: L C ( X, Y )and Hom C (ker T , coker T ) both have natural complex structures if T ∈ Fred C ( X, Y ), and werequire d Φ( T ) to commute with them for all T ∈ O . ectures on Symplectic Field Theory For real-linear operators of index 0, one can use Prop. 3.5 to define the following“relative” invariant. Given two Banach space isomorphisms T ± : X → Y that lie inthe same connected component of Fred R ( X, Y ), define µ spec Z ( T − , T + ) ∈ Z as the parity of the number of times that a generic smooth path [ − , → Fred R ( X, Y )from T − to T + passes through operators with nontrivial kernel. This is welldefined due to the following consequences of standard transversality theory (seeExercise 3.6): first, generic paths { T ( t ) ∈ Fred R ( X, Y ) } t ∈ [ − , are transverse toFred ,k R ( X, Y ) for every k ∈ N , which implies via the codimension formula in Prop. 3.5that they never intersect Fred ,k R ( X, Y ) for k ≥
2, and their intersections withFred , R ( X, Y ) are transverse and thus isolated. Second, transversality also holdsfor generic homotopies[0 , × [ − , → Fred R ( X, Y ) : ( s, t ) T s ( t )with fixed end points between any pair of generic paths T ( t ) and T ( t ), so that theset of intersections with Fred ,k R ( X, Y ) is again empty for k ≥ , × [ − ,
1] for k = 1. This submanifold, moreover,is disjoint from [0 , × {− , } since T s ( ±
1) = T ± , and it is also compact sincethe set of T ∈ Fred R ( X, Y ) with nontrivial kernel is a closed subset. We thereforeobtain a compact 1-dimensional cobordism between the intersection sets of T and T respectively with Fred , R ( X, Y ), implying that the count of intersections modulo 2does not depend on the choice of generic path.
Exercise . Convince yourself that the standard results (as in e.g. [
Hir94 , § f : M → N and submanifolds A ⊂ N continue to hold—with minimal modifications to theproofs—when N is an infinite-dimensional Banach manifold and A ⊂ N has finitecodimension. Exercise . For matrices A ± ∈ GL( n, R ), show that µ spec Z ( A − , A + ) = 0 if andonly if det A + and det A − have the same sign. We now add the following as-sumptions to the setup from the previous subsection: • Y is a Hilbert space H over F , with inner product denoted by h , i H ; • X is an F -linear subspace D ⊂ H , carrying a Banach space structure forwhich the inclusion D ֒ → H is a compact linear operator.The notation D = X is motivated by the fact that if T ∈ L F ( D , H ), then we canalso regard T as an unbounded operator on H with domain D and thus considerthe spectrum of T , see § H is a Hilbert space, the space L F ( H ) of bounded linear operators from H to itself contains a distinguished closed linear subspace L sym F ( H ) ⊂ L F ( H ) , consisting of self-adjoint operators. For operators that are bounded from D to H but not necessarily defined or bounded on H , there is also the space of symmetric Chris Wendl operators L sym F ( D , H ) := (cid:8) T ∈ L F ( D , H ) (cid:12)(cid:12) h x, T y i H = h T x, y i H for all x, y ∈ D (cid:9) . Important examples of symmetric operators are those which are self-adjoint (seeRemark 3.11 below), though for our purposes, it will suffice to restrict attention tosymmetric operators that are also Fredholm with index 0. It turns out that the spaceof symmetric operators in Fred , F ( D , H ) is a canonically co-oriented hypersurface in L sym F ( D , H ), so that the invariant µ spec Z ( T − , T + ) defined above has a natural integer-valued lift when T ± are symmetric. We will need a slightly more specialized versionof this statement in order to give a general definition of spectral flow.In the following, we letFred sym F ( D , H ) := Fred F ( D , H ) ∩ L sym F ( D , H )denote the space of symmetric Fredholm operators with index 0, and for k ∈ N ,Fred sym ,k F ( D , H ) := Fred sym F ( D , H ) ∩ Fred ,k F ( D , H ) . Given T ref ∈ Fred sym F ( D , H ), consider the spaceFred sym F ( D , H , T ref ) := (cid:8) T ref + K : D → H (cid:12)(cid:12) K ∈ L sym F ( H ) (cid:9) . Note that the restriction of each K ∈ L F ( H ) to D is a compact operator D → H ,thus Fred sym F ( D , H , T ref ) has a natural continuous inclusion into Fred sym R ( D , H ). It isalso an affine space over L sym F ( H ) and can thus be regarded naturally as a smoothBanach manifold locally modeled on L sym F ( H ); in particular, its tangent spaces are T T (Fred sym F ( D , H , T ref )) = L sym F ( H ) . A remark about the case F = C is in order: L sym C ( D , H ) is a real -linear and not acomplex subspace of L C ( D , H ), thus Fred sym C ( D , H , T ref ) is a real Banach manifoldbut does not carry a natural complex structure. Lemma . For any T ∈ L sym F ( D , H ) that is Fredholm with index , ker T is theorthogonal complement of im T in H , hence there exist splittings into closed linearsubspaces D = V ⊕ K, H = W ⊕ C where K = C = ker T , W = im T and V = W ∩ D . Proof. If x ∈ K := ker T , then symmetry implies h x, T y i H = h T x, y i H = 0for all y ∈ D , hence K ⊂ W ⊥ , where W := im T . But since ind T = 0, thedimension of ker T equals the codimension of im T , implying that K already has thelargest possible dimension for a subspace that intersects W trivially, and therefore W ⊕ K = H . Since K is also a subspace of D and the latter is a subspace of H , any x ∈ D can be written uniquely as x = v + k where k ∈ K and v ∈ W ∩ D =: V .The continuous inclusion of D into H and the fact that W is closed in H imply that V is a closed subspace of D . (cid:3) We now have the following modification of Prop. 3.5. ectures on Symplectic Field Theory Proposition . For each integer k ≥ , the subset Fred sym ,k F ( D , H , T ref ) := (cid:8) T ∈ Fred sym F ( D , H , T ref ) (cid:12)(cid:12) dim F ker T = k (cid:9) is a smooth finite-codimensional Banach submanifold of Fred sym F ( D , H , T ref ) , with codim R Fred sym ,k F ( D , H , T ref ) = ( k ( k + 1) / if F = R ,k if F = C . In particular,
Fred sym , F ( D , H , T ref ) is a submanifold of Fred sym F ( D , H , T ref ) with codi-mension , and moreover, it carries a canonical co-orientation. Proof.
Given T ∈ Fred sym ,k F ( D , H , T ref ), fix the splittings D = V ⊕ K and H = W ⊕ K as in Lemma 3.8. Using these in the construction of the map Φfrom (3.5) produces a neighborhood O ⊂
Fred sym F ( D , H , T ref ) of T such that, byLemma 3.4, { T ∈ O | dim F ker T = k } = Φ − (0), whereΦ : O →
End F ( K ) : (cid:18) A BC D (cid:19) D − CA − B . Since the splittings are orthogonal, an element T = (cid:18) A BC D (cid:19) ∈ O is symmetric ifand only if h x, A y i H = h A x, y i H for all x, y ∈ V , h x, D y i H = h D x, y i H for all x, y ∈ K, h x, B y i H = h C x, y i H for all x ∈ V , y ∈ K, h x, C y i H = h B x, y i H for all x ∈ K , y ∈ V , and it follows then that Φ( T ) ∈ End sym F ( K ), where End sym F ( K ) ⊂ End F ( K ) is the realvector space of symmetric (or Hermitian when F = C ) linear maps on ( K, h , i H ).We thus have O ∩
Fred sym ,k F ( D , H , T ref ) = Φ − (0) with Φ regarded as a smooth map O ∩
Fred sym F ( D , H , T ref ) → End sym F ( K ). The derivative at T again takes the form d Φ( T ) : L sym F ( H ) → End sym F ( K ) : (cid:18) A ′ B ′ C ′ D ′ (cid:19) D ′ , where now the block matrix represents an element of L sym F ( H ) with respect to thesplitting H = W ⊕ K . This operator is evidently surjective, hence by the implicitfunction theorem, Φ − (0) is a smooth Banach submanifold with codimension equalto dim R End sym F ( K ).Finally, we observe that in the case k = 1, the above identifies Fred sym , F ( D , H , T ref )locally with the zero set of a submersion to End sym F ( K ), which is a real 1-dimensionalvector space since K is a 1-dimensional vector space over F . The canonical isomor-phism R → End sym F ( K ) : a a thus determines a co-orientation on Fred sym , F ( D , H , T ref ). (cid:3) Chris Wendl
The canonical co-orientation of Fred sym , F ( D , H , T ref ) makes it natural to definesigned intersection numbers between Fred sym , F ( D , H , T ref ) and smooth paths in theambient space Fred sym F ( D , H , T ref ). The codimensions of Fred sym ,k F ( D , H , T ref ) foreach k ≥ Definition . Suppose T + : T − ∈ Fred sym F ( D , H , T ref ) are both Banachspace isomorphisms D → H . The spectral flow µ spec ( T − , T + ) ∈ Z from T − to T + is then defined as the signed count of intersections of T : [ − , → Fred sym F ( D , H , T ref ) with Fred sym , F ( D , H , T ref ), where the latter is assumed to carrythe co-orientation given by Prop. 3.9, and T : [ − , → Fred sym F ( D , H , T ref ) is anysmooth path that is transverse to Fred sym ,k F ( D , H , T ref ) for every k ≥ T ( ±
1) = T ± . Continuing in the setting of the previoussubsection, we shall now regard each T ∈ Fred sym F ( D , H , T ref ) as an unboundedoperator on H with domain D , see e.g. [ RS80 , Chapter VIII]. Notice that for eachscalar λ ∈ F , the operator T − λ also belongs to Fred sym F ( D , H , T ref ). The spectrum σ ( T ) ⊂ F of T is defined as the set of all λ ∈ F for which T − λ : D → H does not ad-mit a bounded inverse. In particular, λ ∈ σ ( T ) is an eigenvalue of T whenever T − λ : D → H has nontrivial kernel, and the dimension of this kernel is called the multiplicity of the eigenvalue. We call λ a simple eigenvalue if it has multiplic-ity 1. By a standard argument familiar to both mathematicians and physicists, theeigenvalues of a symmetric complex-linear operator are always real. Remark . If D ⊂ H is dense, then the adjoint of T is defined as anunbounded operator T ∗ with domain D ∗ satisfying h x, T y i H = h T ∗ x, y i H for all x ∈ D ∗ , y ∈ D , where D ∗ is the set of all x ∈ H such that there exists z ∈ H satisfying h x, T y i H = h z, y i H for all y ∈ D . One says that T is self-adjoint if T = T ∗ , which meansboth that T is symmetric and D = D ∗ . In many applications (e.g. in Exercise 3.29),the latter amounts to a condition on “regularity of weak solutions”. This conditionimplies that the inclusion ker T ֒ → (im T ) ⊥ —valid for all symmetric operators—isalso surjective, so if T : D → H is Fredholm, it is then automatic that ind( T ) = 0. Proposition . Assume T ∈ Fred sym F ( D , H , T ref ) . Then:(1) Every λ ∈ σ ( T ) is an eigenvalue with finite multiplicity.(2) The spectrum σ ( T ) is a discrete subset of R .(3) Suppose λ ∈ σ ( T ) is an eigenvalue with multiplicity m ∈ N and ǫ > ischosen such that no other eigenvalues lie in [ λ − ǫ, λ + ǫ ] . Then T has a ectures on Symplectic Field Theory neighorhood O ⊂
Fred sym F ( D , H , T ref ) such that for all T ∈ O , X λ ∈ σ ( T ) ∩ [ λ − ǫ,λ + ǫ ] m ( λ ) = m, where m ( λ ) ∈ N denotes the multiplicity of λ ∈ σ ( T ) . Proof.
For every λ ∈ F , T − λ is a Fredholm operator with index 0, so it isa Banach space isomorphism D → H and thus has a bounded inverse if and onlyif its kernel is trivial. The Fredholm property also implies that the kernel is finitedimensional whenever it is nontrivial, so this proves (1).For (2) and (3), let us assume F = C , as the case F = R will follow by takingcomplexifications of real vector spaces. We claim therefore that σ ( T ) is a discretesubset of C . To see this, suppose λ ∈ R is an eigenvalue of T with multiplicity m ,so T − λ ∈ Fred sym ,m C ( D , H ) . By Lemma 3.8, there are splittings D = V ⊕ K and H = W ⊕ K with K =ker( T − λ ), W = im( T − λ ) and V = W ∩ D . Any scalar λ ∈ C appears inblock-diagonal form (cid:18) λ λ (cid:19) with respect to these splittings, and the block form for T is thus T = (cid:18) A + λ λ (cid:19) for some Banach space isomorphism A : V → W . Writing nearby operators T ∈ Fred C ( D , H ) as (cid:18) A BC D (cid:19) , we can imitate the construction in (3.5) to pro-duce neighborhoods O ( T ) ⊂ Fred C ( D , H ) of T and D ǫ ( λ ) ⊂ C of λ , admittinga holomorphic mapΦ : O ( T ) × D ǫ ( λ ) → End C ( K ) : ( T , λ ) ( D − λ ) − C ( A − λ ) − B such that ker( T − λ ) ∼ = ker Φ( T , λ ). The set of eigenvalues of T near λ is then thezero set of the holomorphic function(3.6) D ǫ ( λ ) → C : λ det Φ( T , λ ) . This function cannot be identically zero since there are no eigenvalues outside of R ,thus the zero at λ is isolated, proving (2).To prove (3), note finally that if the neighborhood O ( T ) ⊂ Fred C ( D , H ) of T is sufficiently small, then for every T ∈ O ( T ), the holomorphic function f T : D ǫ ( λ ) → C : λ det Φ( T , λ )has the same algebraic count of zeroes in D ǫ ( λ ), all of which lie in [ λ − ǫ, λ + ǫ ]if T is symmetric. Observe moreover that since ∂ λ Φ( T , λ ) = − ∈ End C ( K ) , we are free to assume after possibly shrinking ǫ and O ( T ) that ∂ λ Φ( T , λ ) is alwaysa nonsingular transformation in End C ( K ). Since Φ( T , λ ) is in End sym C ( K ) and thusdiagonalizable whenever T is symmetric and λ ∈ R , it follows via Exercise 3.13 Chris Wendl below that the order of any zero f T ( λ ) = 0 is precisely the multiplicity of λ as aneigenvalue of T . (cid:3) Exercise . Suppose
U ⊂ C is an open subset, A : U → C n × n is a holomor-phic map and z ∈ U is a point at which A ( z ) is noninvertible but diagonalizable,and A ′ ( z ) ∈ GL( n, C ). Show that dim C ker A ( z ) is the order of the zero of theholomorphic function det A : U → C at z .The next result implies that for a generic path of symmetric index 0 operatorsas appears in our definition of µ spec ( T − , T + ), the spectral flow is indeed a signedcount of eigenvalues crossing 0. Proposition . Suppose { T t ∈ Fred sym F ( D , H , T ref ) } t ∈ ( − , is a smooth pathand λ ∈ R is a simple eigenvalue of T . Then:(1) For sufficiently small ǫ > , there exists a unique smooth function λ :( − ǫ, ǫ ) → R such that λ (0) = λ and λ ( t ) is a simple eigenvalue of T t foreach t ∈ ( − ǫ, ǫ ) .(2) The derivative λ ′ (0) is nonzero if and only if the intersection of the path { T t − λ ∈ Fred sym F ( D , H , T ref ) } t ∈ ( − , with Fred sym , F ( D , H , T ref ) at t = 0 is transverse, and the sign of λ ′ (0) is then the sign of the intersection. Proof.
Using the same construction as in the proof of Proposition 3.12, we canfind small numbers ǫ > δ > (cid:8) ( t, λ ) ∈ ( − ǫ, ǫ ) × ( λ − δ, λ + δ ) (cid:12)(cid:12) λ ∈ σ ( T t ) (cid:9) = Φ − (0) , whereΦ : ( − ǫ, ǫ ) × ( λ − δ, λ + δ ) → End sym F ( K ) : ( t, λ ) ( D t − λ ) − C t ( A t − λ ) − B t , and we write T t = (cid:18) A t B t C t D t (cid:19) with respect to splittings D = V ⊕ K and H = W ⊕ K with K = ker( T − λ ), W = im( T − λ ) and V = W ∩ D . In saying this, we’veimplicitly used the assumption that λ is a simple eigenvalue, as it follows thatdim F ker( T − λ ) cannot be larger than 1 for any T near T and λ near λ , so thatΦ − (0) catches all nearby eigenvalues. Simplicity also means that End sym F ( K ) is real1-dimensional, and we have ∂ t Φ(0 , λ ) = ∂ t D t | t =0 , ∂ λ Φ(0 , λ ) = − . The implicit function theorem thus gives Φ − (0) near (0 , λ ) the structure of asmooth 1-manifold with tangent space at (0 , λ ) spanned by the vector ∂ t + ( ∂ t D t | t =0 ) ∂ λ , where we are identifying ∂ t D t | t =0 ∈ End sym F ( K ) with a real number via the naturalisomorphism End sym F ( K ) = R . Therefore Φ − (0) can be written as the graph ofa uniquely determined smooth function λ , whose derivative at zero is a multipleof ∂ t D t | t =0 . This proves both statements in the proposition, since by the proof ofProposition 3.9, the intersection of { T t } t ∈ ( − , with Fred sym , F ( D , H , T ref ) is trans-verse if and only if ∂ t D t | t =0 = 0, and its sign is then the sign of ∂ t D t | t =0 . (cid:3) ectures on Symplectic Field Theory The purpose of the next lemma is to prevent eigenvalues from escaping to ±∞ under smooth families of operators in Fred sym F ( D , H , T ref ). Lemma . Suppose { K t ∈ L sym F ( H ) } t ∈ ( a,b ) is a smooth path of symmetricbounded linear operators, and λ : ( a, b ) → R is a smooth function such that for every t ∈ ( a, b ) , λ ( t ) is a simple eigenvalue of T t := T ref + K t ∈ Fred sym F ( D , H , T ref ) . Then | ˙ λ ( t ) | ≤ k ∂ t K t k L ( H ) for all t ∈ ( a, b ) . Proof.
Since { T t − λ ( t ) ∈ Fred sym F ( D , H , T ref ) } t ∈ ( a,b ) is a smooth family ofoperators in Fred F ( D , H ) with 1-dimensional kernel, one can use the local familiesof isomorphisms Ψ( T t − λ ( t )) ∈ L F ( D ) from Lemma 3.4 to find a smooth family ofeigenvectors x ( t ) ∈ ker( T t − λ ( t )) for t ∈ ( a, b ). Normalize these so that k x ( t ) k H =1 for all t . Then 0 = ∂ t h x ( t ) , x ( t ) i H = h ˙ x ( t ) , x ( t ) i H + h x ( t ) , ˙ x ( t ) i H and λ ( t ) = h x ( t ) , T t x ( t ) i H , so writing ˙ K t := ∂ t K t = ∂ t T t , we have˙ λ ( t ) = ∂ t h x ( t ) , T t x ( t ) i H = h x ( t ) , ˙ K t x ( t ) i H + h ˙ x ( t ) , T t x ( t ) i H + h x ( t ) , T t ˙ x ( t ) i H = h x ( t ) , ˙ K t x ( t ) i H , as the last two terms in the first line become λ ( t ) [ h ˙ x ( t ) , x ( t ) i H + h x ( t ) , ˙ x ( t ) i H ] = 0since T t is symmetric and T t x ( t ) = λ ( t ) x ( t ). We obtain | ˙ λ ( t ) | ≤ k x ( t ) k H k ˙ K t k L ( H ) k x ( t ) k H = k ˙ K t k L ( H ) . (cid:3) Specializing further, we now set H and D equal to the specific real Hilbert spaces H := L ( S , R n ) , D := H ( S , R n ) , and set T ref := − J ∂ t , where J denotes the standard complex structure on R n = C n . Observe that any bounded linear operator on L determines a compact operator H → L via composition with the compact inclusion. In particular, we shallconsider compact perturbations of − J ∂ t in the form(3.7) A = − J ∂ t − S ( t )with S : S → End sym R ( R n ) smooth. It is straightforward to check that this operatoris symmetric with respect to the L -product since S ( t ) is symmetric for every t . Thefollowing then implies that A ∈ Fred sym R ( D , H , T ref ). Lemma . The operator − J ∂ t : H ( S , R n ) → L ( S , R n ) is Fredholm withindex . Proof.
Since J defines an isomorphism, it suffices actually to show that theordinary differential operator ∂ t : H ( S , R n ) → L ( S , R n )is Fredholm with index 0. The kernel of this operator is the space of constantfunctions S → R n , which has dimension 2 n . To compute the dimension of thecokernel, we observe that if f = ∂ t F lies in the image of this operator, we have R S f ( t ) dt = 0 since F is periodic in t . Conversely, if R S f ( t ) dt = 0 with f ∈ Chris Wendl L ( S , R n ), then the function F ( s ) = R s f ( t ) dt is periodic in s and defines anelement of H ( S , R n ) satisfying ∂ t F = f . Hence the image of ∂ t is exactly the setim( ∂ t ) = (cid:26) f ∈ L ( S , R n ) (cid:12)(cid:12)(cid:12)(cid:12) Z S f ( t ) dt = 0 (cid:27) , which has codimension 2 n . (cid:3) The proof of Theorem 3.3 requires only one more technical ingredient, whoseproof is given in Appendix C and should probably be skipped on first reading unlessyou have already read Lecture 7 or seen similar applications of the Sard-Smaletheorem. You might however find the result plausible in accordance with the notionthat maps from 2-dimensional domains, such as a map of the form( − , × R → Fred sym R ( D , H , T ref ) : ( t, λ ) T t − λ should generically not intersect submanifolds that have codimension 3 or more, suchas Fred sym ,k R ( D , H , T ref ) when k ≥ Lemma . Fix a smooth map S : [ − , × S → End sym R ( R n ) and considerthe -parameter family of unbounded linear operators A s := − J ∂ t − S ( s, · ) : L ( S , R n ) ⊃ H ( S , R n ) → L ( S , R n ) for s ∈ [ − , . One can arrange after a C ∞ -small perturbation of S fixed at s = ± that the following conditions hold:(1) For each s ∈ ( − , , all eigenvalues of A s are simple.(2) All intersections of the path ( − , → Fred sym R ( D , H , T ref ) : s A s with Fred sym , R ( D , H , T ref ) are transverse. (cid:3) Proof of Theorem 3.3.
Given a smooth family { A s } s ∈ [ − , as stated in thetheorem, use Lemma 3.17 to obtain a C ∞ -small perturbation for which the eigen-values are simple for s ∈ ( − ,
1) and all intersections with Fred sym , R ( D , H ) aretransverse. Proposition 3.14 then implies that the eigenvalues depend smoothlyon s , and Lemma 3.15 imposes a uniform bound on their derivatives with respectto s so that each one varies only in a bounded subset of R for s ∈ ( − , s ∈ ( − ,
1) therefore extend to continuous familiesfor s ∈ [ − ,
1] since the space of noninvertible Fredholm operators with index 0 isclosed. Proposition 3.12 ensures moreover that these continuous families hit everyeigenvalue with the correct multiplicity at s = ±
1, and by Proposition 3.14, theformula for µ spec ( A − , A + ) stated in the theorem is correct for the perturbed familywith simple eigenvalues and transverse crossings. To obtain the same result for theoriginal family, suppose we have a sequence of perturbations { A νs } s ∈ [ − , convergingin C ∞ as ν → ∞ to { A s } s ∈ [ − , . Lemma 3.15 then provides a uniform C -boundfor each sequence of smooth families of eigenvalues, so they have C -convergent sub-sequences as ν → ∞ , giving rise to the continuous families in the statement of thetheorem. (cid:3) ectures on Symplectic Field Theory Remark . It is important to understand that the definition of spectral flowdepends on the particular co-orientation of Fred sym , F ( D , H , T ref ) that arose in theproof of Prop. 3.9; we saw in Prop. 3.14 that this is indeed the right co-orientationto use if we want to interpret signed intersections with Fred sym , F ( D , H , T ref ) assigned crossing numbers of eigenvalues. In the non-symmetric setting of § , R ( X, Y ) is also co-orientable; this is obvious in the finite-dimensional case since Fred , R ( R n , R n ) is then a regular level set of the determinantfunction. Moreover, Fred , R ( R n , R n ) is connected (see Exercise 3.19 below), so theco-orientation is unique up to a sign. One can therefore lift the Z -valued spectralflow of § Z , but as in Exercise 3.7, the result will be a different and muchless interesting invariant than µ spec ( A − , A + ), as its value will always be either 0 (ifdet A − and det A + have the same sign) or ± sym , R ( D , H , T ref ) must gen-erally differ on some connected components from any possible co-orientation of thelarger hypersurface Fred , R ( D , H ) ⊂ Fred R ( D , H ). Exercise . Show that the space Fred , R ( R , R ) of rank 1 matrices in R × isconnected, but the space Fred sym , R ( R , R ) of symmetric rank 1 matrices is not, andthat the canonical co-orientation of Fred sym , R ( R , R ) coming from Prop. 3.9 differson some components from any possible co-orientation of Fred , R ( R , R ) ⊂ R × . Hint: A non-symmetric -by- matrix may have rank even if both of its eigenvaluesare . For symmetric matrices this cannot happen. Exercise . Find a smooth path A : [ − , → R × of symmetric matricessuch that A ± := A ( ±
1) are both invertible and µ spec ( A − , A + ) = 2, but A + and A − can also be connected by a smooth path of (not necessarily symmetric) invertiblematrices in R × . Before returning to contact geometry, let’s quickly revisit the Floer homology fora time-dependent Hamiltonian { H t : M → R } t ∈ S on a symplectic manifold ( M, ω ).In Lecture 1, we introduced the symplectic action functional A H : C ∞ contr ( S , M ) → R and wrote down the formula ∇A H ( γ ) = J t ( γ ) ( ˙ γ − X t ( γ )) ∈ Γ( γ ∗ T M ) =: T γ C ∞ contr ( S , M )for the “unregularized” gradient of A H at a contractible loop γ ∈ C ∞ contr ( S , M ).Here X t denotes the Hamiltonian vector field and J t is a time-dependent family ofcompatible almost complex structures, which determines the L -product h η , η i L = Z S ω ( η ( t ) , J t η ( t )) dt. The critical points of A H are the loops γ such that ∇A H ( γ ) = 0. Formally, theHessian of A H at γ ∈ Crit( A H ) is the “linearization of A H at γ ,” which gives alinear operator A γ := ∇ A H ( γ ) : Γ( γ ∗ T M ) → Γ( γ ∗ T M ) . Chris Wendl
To write it down, one can choose any connection ∇ on M , and choose for η ∈ Γ( γ ∗ T M ) a smooth family { γ ρ : S → M } ρ ∈ ( − ǫ,ǫ ) with γ = γ and ∂ ρ γ ρ | ρ =0 = η , andthen compute A γ η := ∇ ρ [ ∇A H ( γ ρ )] | ρ =0 . The result is independent of the choice of connection since ∇A H ( γ ) = 0. Exercise . Show that if the connection ∇ on M is chosen to be symmetric,then A γ η = J t ( ∇ t η − ∇ η X t ).We now introduce the class of symmetric operators that appear in asymptoticformulas in SFT. Fix a (2 n − M, ξ ) with contactform α , induced Reeb vector field R α , and a complex structure J : ξ → ξ compatiblewith the symplectic structure dα | ξ . Let π ξ : T M → ξ denote the projection along R α . The contact action functional is defined by A α : C ∞ ( S , M ) → R : γ Z S γ ∗ α. The first variation of this functional for γ ∈ C ∞ ( S , M ) and η ∈ Γ( γ ∗ T M ) is d A α ( γ ) η = Z S dα ( η, ˙ γ ) dt = − Z S dα ( π ξ ˙ γ, η ) dt. The functional has a built-in degeneracy since it is parametrization-invariant; inparticular, d A α ( γ ) η = 0 whenever η points in the direction of the Reeb vector field,a symptom of the fact that closed Reeb orbits always come in families related toeach other by reparametrization. A loop γ : S → M is critical for A α if andonly if ˙ γ is everywhere tangent to R α , allowing for an infinite-dimensional familyof distinct perturbations—however, there exist preferred parametrizations, namelythose for which ˙ γ is a constant multiple of R α , meaning(3.8) ˙ γ = T · R α ( γ ) , T := A α ( γ ) . Such a loop corresponds to a T -periodic solution x : R → M to ˙ x = R α ( x ), where γ ( t ) = x ( T t ).The discussion above indicates that we cannot derive a “Hessian” of A α in thesame straightforward way as in Floer homology, as the resulting operator will alwayshave nontrivial kernel due to the degeneracy in the R α direction. To avoid this, weshall consider only preferred parametrizations γ : S → M of the form (3.8), andperturbations in directions tangent to ξ , which is transverse to every Reeb orbit.For η ∈ Γ( γ ∗ ξ ), we then have d A α ( γ ) η = Z S dα ( − J π ξ ˙ γ, J η ) dt = h− J π ξ ˙ γ, η i L , where we define an L -product for sections of γ ∗ ξ by(3.9) h η, η ′ i L := Z S dα ( η, J η ′ ) dt. ectures on Symplectic Field Theory It therefore seems sensible to write ∇A α ( γ ) := − J π ξ ˙ γ ∈ Γ( γ ∗ ξ ) , and we shall define the Hessian at a critical point γ as the linearization in ξ directions,i.e. ∇ A α ( γ ) : Γ( γ ∗ ξ ) → Γ( γ ∗ ξ ) . Given η ∈ Γ( γ ∗ ξ ), choose a smooth family { γ ρ : S → M } ρ ∈ ( − ǫ,ǫ ) with γ = γ and ∂ ρ γ ρ | ρ =0 = η , and fix a symmetric connection ∇ on M . Since π ξ ˙ γ = 0, the covariantderivative of ∇A α ( γ ρ ) at ρ = 0 is then ∇ ρ ( − J π ξ ˙ γ ρ ) | ρ =0 = − J ∇ ρ ( π ξ ˙ γ ρ ) | ρ =0 = − J ∇ ρ [ ˙ γ ρ − α ( ˙ γ ρ ) R α ( γ ρ )] | ρ =0 = − J ( ∇ t η − T ∇ η R α − ∂ ρ [ α ( ˙ γ ρ )] | ρ =0 · R α ( γ )) . In the last term, we can write ∂ ρ [ α ( ˙ γ ρ )] | ρ =0 = dα ( η, ˙ γ ) + ∂ t [ α ( η )] = 0 since ˙ γ = T R α ( γ ) and α ( η ) = 0 for η ∈ Γ( γ ∗ ξ ). One can now check that the remaining termsdefine a section of γ ∗ ξ , thus we are led to the following definition. Definition . Given a loop γ : S → M parametrizing a closed Reeb orbitin ( M, ξ = ker α ) with period T ≡ α ( ˙ γ ), the asymptotic operator associated to γ is the first-order differential operator on γ ∗ ξ defined by A γ : Γ( γ ∗ ξ ) → Γ( γ ∗ ξ ) : η
7→ − J ( ∇ t η − T ∇ η R α ) Exercise . Show that A γ is symmetric with respect to the L inner product(3.9) on Γ( γ ∗ ξ ). Moreover, γ is nondegenerate (see § A γ istrivial. Hint for nondegeneracy: Consider the pullback of γ ∗ ξ via the cover R → S = R / Z , and show that solutions to ∇ t η − T ∇ η R α = 0 on the pullback are givenby operating on ξ γ (0) with the linearized Reeb flow. To see this, try differentiatingfamilies of solutions to the equation ˙ x = T R α ( x ) . Remark . Another way of phrasing the hint in the the above exercise isas follows: A γ can also be written as − J b ∇ t , where b ∇ t is the unique symplecticconnection on ( γ ∗ ξ, dα ) for which parallel transport is given by the linearized Reebflow.You might be slightly concerned about the sign difference between the two for-mulas we’ve derived for asymptotic operators in contact geometry and in Floerhomology. I also find this troubling, but the discrepancy seems to originate fromthe fact that our account of Floer homology has referred always to the negative gra-dient flow of A H , while SFT is actually defined via the positive gradient flow of A α .The words “gradient flow” in SFT must in any case be interpreted very loosely. If u : [0 , ∞ ) × S → R × M is the cylindrical end of a finite-energy J -holomorphic curve for some J ∈ J ( α ) aswe described in Lecture 1, then u ( s, t ) does not satisfy anything so straightforwardas ∂ s − ∇A α ( u ( s, · )) = 0, but it does satisfy π ξ ∂ s u + J π ξ ∂ t u = 0 , which can be interpreted as the projection of a positive gradient flow equation to thecontact bundle. This observation is a local symptom of a more important global fact Chris Wendl that follows from Stokes’ theorem: any asymptotically cylindrical J -holomorphiccurve u : ˙Σ → R × M with positive and negative punctures Γ ± asymptotic to orbits { γ z } z ∈ Γ ± satisfies X z ∈ Γ + A α ( γ ) − X z ∈ Γ − A α ( γ ) = Z ˙Σ u ∗ dα ≥ . This generalizes the basic fact in Floer homology that flow lines decrease action and,conversely, have their energy controlled by the action.We would now like to develop some of the general properties of asymptoticoperators. Recall that on any symplectic vector bundle (
E, ω ), a compatible complexstructure J determines a Hermitian inner product h v, w i = ω ( v, J w ) + iω ( v, w ) , and conversely, any Hermitian inner product on a complex vector bundle determinesa symplectic structure via the same relation. For this reason, we shall refer to anyvector bundle E with a compatible pair ( J, ω ) as a
Hermitian vector bundle . A unitary trivialization of such a bundle is a trivialization that identifies fibers with R n such that J and ω become the standard complex structure J and symplecticstructure ω respectively. Definition . Fix a Hermitian vector bundle (
E, J, ω ) over S . An asymp-totic operator on ( E, J, ω ) is any real-linear differential operator A : Γ( E ) → Γ( E )that takes the form(3.10) A : C ∞ ( S , R n ) → C ∞ ( S , R n ) : η
7→ − J ∂ t η − S ( t ) η in unitary trivializations, where S : S → End( R n ) is a smooth loop of symmetricmatrices.Equivalently, an asymptotic operator on ( E, J, ω ) is any operator of the form − J ∇ where ∇ is a symplectic connection on E . Exercise . Show that any asymptotic operator on a Hermitian vector bun-dle (
E, J, ω ) over S is symmetric with respect to the real L bundle metric h η , η i L := Z S ω ( η ( t ) , J η ( t )) dt. Exercise . Show that the asymptotic operator A γ for a closed Reeb orbit γ is also an asymptotic operator on ( γ ∗ ξ, J, dα ) in the sense of Definition 3.25.For functional analytic purposes, we shall regard asymptotic operators on Her-mitian bundles ( E, J, ω ) as bounded real-linear operators A : H ( E ) → L ( E ) . By Lemma 3.16, all asymptotic operators are then Fredholm with index 0, and anytwo such operators on the same bundle are compact perturbations of each other.Regarding them alternatively as unbounded symmetric operators on L ( E ), thespectral flow µ spec ( A − , A + ) ∈ Z ectures on Symplectic Field Theory between two such operators A ± with trivial kernel is defined by choosing any unitarytrivialization to write both in the form − J ∂ t − S ( t ), and it is independent of thischoice. The following is what we mean when we say that critical points of the actionfunctional have “infinite Morse index” and “infinite Morse co-index”: Proposition . Every asymptotic operator has infinitely many eigenvaluesof both signs.
Proof.
It is easy to verify that this is true for A := − J ∂ t : H ( S , R n ) → L ( S , R n ); see the proof of theorem 3.35 below. It is therefore also true for A + ǫ for any ǫ ∈ R , and this operator has trivial kernel whenever ǫ π Z . For anyother trivialized asymptotic operator A with 0 σ ( A ), the result then follows fromTheorem 3.3 since µ spec ( A + ǫ, A ) is finite, and this is precisely the signed countof eigenvalues which change sign. The condition 0 σ ( A ) can then be lifted byreplacing A with A + ǫ . (cid:3) Exercise . Show that asymptotic operators are self-adjoint (as unboundedoperators on L with domain H ) in the sense of Remark 3.11. We are now in a position to define a suitable replacement for the Morse index inthe context of SFT. We shall say that an asymptotic operator A is nondegenerate whenever 0 σ ( A ). We will begin by defining the Conley-Zehnder index as aninteger-valued invariant of homotopy classes of nondegenerate asymptotic operatorson the trivial Hermitian bundle S × R n ; the definition on arbitrary Hermitianbundles will then depend on a choice of trivialization.It is customary elsewhere in the literature (see e.g. [ SZ92 ]) to adopt a somewhatdifferent perspective on the Conley-Zehnder index, in which it defines an integer-valued invariant of connected components of the space of “nondegenerate symplecticarcs” (cid:8) Ψ ∈ C ([0 , , Sp(2 n )) (cid:12)(cid:12) Ψ(0) = and 1 σ (Ψ(1)) (cid:9) . These are two different perspectives on the same notion. A dictionary from oursto the other perspective is provided by associating to any trivialized nondegenerateasymptotic operator A = − J ∂ t − S ( t ) the symplectic arc Ψ defined by the initialvalue problem ( − J ∂ t − S ( t ))Ψ( t ) = 0 , Ψ(0) = . Conversely, any smooth symplectic arc determines via this same formula a smoothpath of symmetric matrices S : [0 , → End( R n ), producing a mild generalizationof our notion of an asymptotic operator. If S ( t ) is not continuous on S but is continuous on [0 , − J ∂ t − S ( t ) cannot beregarded as a linear operator on C ∞ ( S , R n ) but is still a very well-behaved symmetric Fredholmoperator from H ( S ) to L ( S ). All of the important functional analytic results in this lecturecan thus be generalized to allow this. Chris Wendl
Definition . The
Conley-Zehnder index associates to every trivializednondegenerate asymptotic operator A : H ( S , R n ) → L ( S , R n ) as in (3.10) aninteger µ CZ ( A ) ∈ Z determined uniquely by the following properties:(1) Set µ CZ ( A ) := 0 for the operator A = − J ∂ t − (cid:18) − (cid:19) .(2) For any two nondegenerate operators A ± , set µ CZ ( A − ) − µ CZ ( A + ) := µ spec ( A − , A + ) . Definition . Given a nondegenerate asymptotic operator A on a Hermitianbundle ( E, J, ω ) over S and a choice of complex trivialization τ for ( E, J ), the
Conley-Zehnder index of A with respect to τ is the integer µ τ CZ ( A ) ∈ Z defined by choosing any unitary trivialization homotopic to τ to write A as anoperator H ( S , R n ) → L ( S , R n ) and then plugging in Definition 3.30.If γ is a nondegenerate Reeb orbit γ in a (2 n − M, ξ = ker α ), then for any complex trivialization τ of γ ∗ ξ → S , the Conley-Zehnder index of γ relative to τ is defined as µ τ CZ ( γ ) := µ τ CZ ( A γ ) . Remark . From the perspective of [
SZ92 ], µ τ CZ ( γ ) is the Conley-Zehnderindex of the linearized Reeb flow along γ restricted to ξ , expressed via a choice ofunitary trivialization as a nondegenerate arc in Sp(2 n − Exercise . Show that if A and A are nondegenerate asymptotic operatorson Hermitian bundles E and E respectively, then A ⊕ A defines a nondegenerateasymptotic operator on E ⊕ E , and given trivializations τ j for j = 1 , µ τ ⊕ τ CZ ( A ⊕ A ) = µ τ CZ ( A ) + µ τ CZ ( A ) . The following is a functional-analytic version of the well-known fact that theConley-Zehnder index classifies homotopy classes of nondegenerate symplectic arcs.
Theorem . On any Hermitian bundle ( E, J, ω ) → S with complex trivial-ization τ , two nondegenerate asymptotic operators A ± lie in the same connected com-ponent of the space of nondegenerate asymptotic operators if and only if µ τ CZ ( A + ) = µ τ CZ ( A − ) . Proof.
Trivializing the bundle, we need to show that if A ± = − J ∂ t − S ± ( t )satisfy µ spec ( A − , A + ) = 0, then there exists a path of asymptotic operators betweenthem for which no eigenvalues cross 0. To see this, we can first choose any path { A t } t ∈ [ − , of asymptotic operators with A ± = A ± , and then use Lemma 3.17 toadd generic compact perturbations producing a family (cid:8) A ′ t ∈ Fred sym R ( H , L , A + ) (cid:9) t ∈ [ − , ectures on Symplectic Field Theory whose intersections with Fred sym ,k R ( H , L , A + ) are transverse for every k ≥
1, henceonly simple eigenvalues cross 0 and they cross transversely. Any neighboring pairof crossings with opposite signs can then be eliminated by changing { A ′ t } t ∈ [ − , to { A ′ t + c ( t ) } t ∈ [ − , for a suitable choice of smooth function c : [ − , → R . Sincethe spectral flow is zero, one can repeat this modification until one obtains a pathof perturbed operators with no crossings, and it is a small perturbation of the pathof asymptotic operators { A t + c ( t ) } t ∈ [ − , . Since A ± are both nondegenerate, onecan assume moreover that all eigenvalues of A t + c ( t ) stay a fixed distance δ > δ is independent of the perturbation. One can therefore “turnoff the perturbation” as in the proof of Theorem 3.3, i.e. there exists a sequence ofperturbed paths { A νt } t ∈ [ − , converging to { A t + c ( t ) } whose eigenvalues stay a fixeddistance away from 0, and the same is therefore true for the continuous families ofeigenvalues of A t + c ( t ) obtained as ν → ∞ . (cid:3) To compute Conley-Zehnder indices, Exercise 3.33 shows that it suffices if weknow how to compute them for operators on Hermitian line bundles. The next twotheorems provide a tool for handling the latter.
Theorem . Let A = − J ∂ t − S ( t ) : H ( S , R ) → L ( S , R ) , where S ( t ) is a smooth loop of symmetric -by- matrices. For each λ ∈ σ ( A ) , denote thecorresponding eigenspace by E λ ⊂ H ( S , R ) .(1) Every nontrivial eigenfunction e λ ∈ E λ is nowhere zero and thus has awell-defined winding number wind( e λ ) ∈ Z .(2) Any two nontrivial eigenfunctions in the same eigenspace E λ have the samewinding number.(3) If λ, µ ∈ σ ( A ) satisfy λ < µ , then any two nontrivial eigenfunctions e λ ∈ E λ and e µ ∈ E µ satisfy wind( e λ ) ≤ wind( e µ ) .(4) For every k ∈ Z , A has exactly two eigenvalues (counting multiplicity) forwhich the corresponding eigenfunctions have winding number equal to k . Proof.
We follow the proof given in [
HWZ95 ].Observe first that (1) follows from the fact that nontrivial eigenfunctions are so-lutions to an ODE, for which classical existence and uniqueness results are available.Since the trivial map is a solution, every eigenfunction which vanishes at a pointmust be itself trivial, by uniqueness.To prove (2), let ν and ν be nontrivial eigenfunctions for the same eigenvalue λ .If their winding numbers are different, then there exists t ∈ S at which ν ( t ) is anonzero real multiple of ν ( t ), so after rescaling, we can assume ν ( t ) = ν ( t ). But ν and ν are both solutions to the same linear ODE, so this implies ν ( t ) = ν ( t )for all t and thus contradicts the assumption on the winding numbers.We first prove the rest for the case S = 0 and the operator A = − J ∂ t . Given ν ∈ H ( S , R ), written as ν ( t ) = ( x ( t ) , y ( t )), we have that ν is an element of E λ for the operator A if and only ( ˙ y, − ˙ x ) = λ ( x, y ). This has solutions of the form (cid:26) x ( t ) = A cos( λt ) − B sin( λt ) y ( t ) = B cos( λt ) + A sin( λt ) , Chris Wendl for some constants
A, B ∈ R , which are defined on S as long as λ ∈ π Z . In otherwords, the spectrum of this operator is σ ( A ) = 2 π Z . Hence ν ( t ) = ν (0) e iλt , whichhas winding number wind( ν ) = λ π Statements (2) and (3) are now obvious, and (4) follows from the observation that E λ is two-dimensional, so in this case each eigenvalue is to be counted with multiplicitytwo.For the general case, consider the path of asymptotic operators given by { A τ = − J ∂ t − τ S ( t ) } τ ∈ [0 , . Theorem 3.3 gives continuous families { λ j : [0 , → R } j ∈ Z and { ν j : [0 , → H ( S , R ) } j ∈ Z such that for every τ ∈ [0 , ν j ( τ ) is an eigenfunction for theoperator A τ with eigenvalue λ j ( τ ), whose multiplicity is given by the number of i ∈ Z for which λ i ( τ ) = λ j ( τ ), and such that λ n + k (0) = 2 πn , for k = 0 , ν n + k ( τ )) = wind( ν n + k (0)) = n, for k = 0 ,
1. Moreover, since the winding only depends on the eigenvalue, the onlypaths that can possibly meet are λ n and λ n +1 , which implies that the multiplicityof every eigenvalue λ i ( τ ) is at most two, with equality where these two “branches”meet. Hence (3) and (4) follow, where equality in (3) holds if and only if the twobranches of paths of eigenvalues with same winding number end up at differentpoints. (cid:3) The theorem implies the existence of a well-defined and nondecreasing function σ ( A ) → Z : λ wind( λ ) , where wind( λ ) is defined as wind( e λ ) for any nontrivial e λ ∈ E λ , and this functionattains every value exactly twice (counting multiplicity of eigenvalues). Since eigen-values of A are isolated, we can therefore associate to any nondegenerate asymptoticoperator A on the trivial Hermitian line bundle its extremal winding numbers and its parity , α + ( A ) = min λ ∈ σ ( A ) ∩ (0 , ∞ ) wind( λ ) ∈ Z ,α − ( A ) = max λ ∈ σ ( A ) ∩ ( −∞ , wind( λ ) ∈ Z ,p ( A ) = α + ( A ) − α − ( A ) ∈ { , } . (3.11) Theorem . If A is a nondegenerate asymptotic operator on the trivial Her-mitian line bundle S × R → S , then µ CZ ( A ) = 2 α − ( A ) + p ( A ) = 2 α + ( A ) − p ( A ) . ectures on Symplectic Field Theory Proof.
The operator A = − J ∂ t − (cid:18) − (cid:19) satisfies µ CZ ( A ) = 0 by def-inition, and it has two constant eigenfunctions with eigenvalues of opposite signs,hence α − ( A ) = α + ( A ) = 0 , consistent with the stated formula. The general case then follows by computingthe spectral flow from A to any other nondegenerate operator A , and observingthat the winding number associated to any continuous family of eigenvalues (as inTheorem 3.3) for a path { A t } t ∈ [ − , of asymptotic operators cannot change. (cid:3) For any Hermitian line bundle (
E, J, ω ) over S with a nondegenerate asymptoticoperator A , we can similarly choose a complex trivialization τ to define the windingnumbers α τ ± ( A ) ∈ Z and parity p ( A ) = α τ + ( A ) − α τ − ( A ) ∈ { , } ; note that thedependence on τ cancels out in the last formula, so that p ( A ) is independent ofchoices. We then can associate to any nondegenerate Reeb orbit γ in a contact3-manifold ( M, ξ = ker α ) with a trivialization τ of γ ∗ ξ the integers α τ ± ( γ ) and p ( γ ),such that µ τ CZ ( γ ) = 2 α τ − ( γ ) + p ( γ ) = 2 α τ + ( γ ) − p ( γ )holds. Exercise . Given a Hermitian vector bundle (
E, J, ω ) → S with two com-plex trivializations τ j : E → S × R n for j = 1 ,
2, denote bydeg( τ ◦ τ − ) ∈ Z the winding number of det g : S → C \ { } , where g : S → GL( n, C ) is thetransition map appearing in the formula τ ◦ τ − ( t, v ) = ( t, g ( t ) v ). Show that forany asymptotic operator A on ( E, J, ω ), µ τ CZ ( A ) = µ τ CZ ( A ) + 2 deg( τ ◦ τ − ) . Exercise 3.37 provides the useful formula µ τ CZ ( γ ) = µ τ CZ ( γ ) + 2 deg( τ ◦ τ − )for any two trivializations τ , τ of ξ along a nondegenerate Reeb orbit γ . In partic-ular, this shows that the parity µ Z CZ ( γ ) := [ µ τ CZ ( γ )] ∈ Z of the orbit does not depend on a choice of trivialization. We sometimes refer to even orbits and odd orbits accordingly. Exercise . Show that if a Reeb orbit γ : S → M in a contact 3-manifold( M, ξ = ker α ) is nondegenerate and has even parity, then the same is true for all ofits multiple covers γ k : S → M : t γ ( kt ) , k ∈ N . ECTURE 4
Fredholm theory with cylindrical endsContents
In this lecture we will study the class of linear Cauchy-Riemann type operatorsthat arise by linearizing the nonlinear equation for moduli spaces in SFT. We sawin the previous lecture that linearizing PDEs over domains with cylindrical endsnaturally leads one to consider certain symmetric asymptotic operators (e.g. theHessian of a Morse function at its critical points), which have trivial kernel if and onlyif a nondegeneracy (i.e. Morse) condition is satisfied. Our goal in this lecture is towrite down the SFT version of this story and show that the linear Cauchy-Riemanntype operators are Fredholm if their asymptotic operators are nondegenerate.
The setup throughout this lecture will be as follows.Assume (Σ , j ) is a closed connected Riemann surface of genus g ≥
0, Γ ⊂ Σ is afinite set partitioned into two subsetsΓ = Γ + ∪ Γ − , and ˙Σ := Σ \ Γ denotes the resulting punctured Riemann surface. We shall fixa choice of holomorphic cylindrical coordinate near each puncture z ∈ Γ ± ,meaning the following. Given R ≥
0, let ( Z R ± , i ) denote the half-cylinders Z R + := [ R, ∞ ) × S , Z R − := ( −∞ , − R ] × S , Z ± := Z ± , with complex structure i∂ s = ∂ t , i∂ t = − ∂ s in coordinates ( s, t ) ∈ R × S . Thestandard half-cylinders Z ± are each biholomorphically equivalent to the punctureddisk ˙ D := D \ { } via the maps ψ ± : Z ± → ˙ D : ( s, t ) e ∓ π ( s + it ) . For z ∈ Γ ± , we choose a closed neighborhood U z ⊂ Σ of z with a biholomorphicmap ϕ z : ( ˙ U z , j ) → ( Z ± , i ) , Chris Wendl where ˙ U z := U z \ { z } , such that ψ ± ◦ ϕ z : ˙ U z → ˙ D extends holomorphically to U z → D with z
0. One can always find such coordinates by choosing holomorphiccoordinates near z . We can thus view the punctured neighborhoods ˙ U z ⊂ ˙Σ as cylindrical ends Z ± .Suppose ( E, J ) is a smooth complex vector bundle of rank m over ( ˙Σ , j ). An asymptotically Hermitian structure on ( E, J ) is a choice of Hermitian vectorbundles ( E z , J z , ω z ) of rank m associated to each puncture z ∈ Γ ± , together withchoices of complex bundle isomorphisms E | ˙ U z → pr ∗ E z covering ϕ z : ˙ U z → Z ± , where pr : Z ± → S denotes the natural projection to the S factor. This isomorphism induces from any unitary trivialization τ of ( E z , J z , ω z )a complex trivialization(4.1) τ : E | ˙ U z → Z ± × R m over the cylindrical end, which we will call an asymptotic trivialization near z .The bundle ( E z , J z , ω z ) will be referred to as the asymptotic bundle associatedto ( E, J ) near z .Fixing asymptotic trivializations near every puncture, we can now define Sobolevspaces of sections of E by W k,p ( E ) := n η ∈ W k,p loc ( E ) (cid:12)(cid:12)(cid:12) η z ∈ W k,p ( ˚ Z ± , R m ) for every z ∈ Γ ± o , where η z : Z ± → R m denotes the expression of η | ˙ U z in terms of the asymptotic trivi-alization, and we use the standard area form ds ∧ dt on Z ± to define the norm. Since S is compact, different choices of asymptotic trivialization give rise to equivalentnorms, however: Exercise . Convince yourself that different choices of asymptotically Her-mitian structure on E → ˙Σ can give rise to inequivalent W k,p -norms.Any linear Cauchy-Riemann type operator on E has as its target the complexvector bundle F := Hom C ( T ˙Σ , E ) , so sections of F are the same thing as E -valued (0 , τ as in (4.1) then also induces a complex trivialization F | ˙ U z → Z ± × R m : λ τ ( λ ( ∂ s )) , where ∂ s is the vector field on ˙ U z arising from its identification with Z ± . Thistrivialization yields a corresponding definition for the Sobolev spaces W k,p ( F ), whichdepend on the asymptotically Hermitian structure of E but not on the choices ofasymptotic trivializations. Having made these choices, a Cauchy-Riemann typeoperator D : Γ( E ) → Γ( F ) always appears over ˙ U z as a linear map on C ∞ ( Z ± , R m )of the form(4.2) D η ( s, t ) = ¯ ∂η ( s, t ) + S ( s, t ) η ( s, t ) , where ¯ ∂ := ∂ s + J ∂ t and S ∈ C ∞ ( Z ± , End( R m )). ectures on Symplectic Field Theory Definition . Suppose A z is an asymptotic operator on ( E z , J z , ω z ) and D isa linear Cauchy-Riemann type operator on ( E, J ). We say that D is asymptotic toA z at z if D appears in the form (4.2) with respect to an asymptotic trivializationnear z , with k S − S ∞ k C k ( Z R ± ) → R → ∞ for all k ∈ N , where S ∞ ( s, t ) := S ∞ ( t ) is a smooth loop of symmetric matricessuch that A z appears in the corresponding unitary trivialization of ( E z , J z , ω z ) as − J ∂ t − S ∞ .Recall that an asymptotic operator is called nondegenerate if 0 is not in itsspectrum, which means it defines an isomorphism H → L . The objective of thislecture will be to prove the following: Theorem . Suppose ( E, J ) is an asymptotically Hermitian vector bundleover ( ˙Σ , j ) , A z is a nondegenerate asymptotic operator on the associated asymptoticbundle ( E z , J z , ω z ) for each z ∈ Γ , and D is a linear Cauchy-Riemann type operatorasymptotic to A z at each puncture z . Then for every k ∈ N and < p < ∞ , D : W k,p ( E ) → W k − ,p ( F ) is Fredholm. Moreover, ind D and ker D are each independent of k and p , the latterbeing a space of smooth sections whose derivatives of all orders decay to at infinity. Remark . The asymptotic decay conditions on S ( s, t ) in Definition 4.2 canbe relaxed at the cost of limiting the range of k ∈ N for which Theorem 4.3 isvalid. To prove that D : W ,p → L p is Fredholm, it suffices to assume S ( s, · ) → S ∞ uniformly as | s | → ∞ .The index of D is determined by a generalization of the Riemann-Roch formulainvolving the Conley-Zehnder indices µ τ CZ ( A z ) that were introduced in the previouslecture. We will postpone serious discussion of the index formula until the nextlecture, but here is the statement: Theorem . In the setting of Theorem 4.3, ind D = mχ ( ˙Σ) + 2 c τ ( E ) + X z ∈ Γ + µ τ CZ ( A z ) − X z ∈ Γ − µ τ CZ ( A z ) , where τ is an arbitrary choice of asymptotic trivializations, c τ ( E ) ∈ Z is the relativefirst Chern number of E with respect to τ , and the sum is independent of this choice. For the rest of this lecture, we maintain as standing assumptions that k ∈ N ,1 < p < ∞ , and D is a linear Cauchy-Riemann type operator on E asymptotic atthe punctures to a fixed set of asymptotic operators { A z } z ∈ Γ . We will not alwaysneed to assume that the A z are nondegenerate, so this condition will be specifiedwhenever it is relevant. For subdomains Σ ⊂ ˙Σ, we will sometimes denote the W k,p -norm on sections of E restricted to Σ by k η k W k,p (Σ ) := k η k W k,p ( E | Σ0 ) , Chris Wendl and we will use the same notation for sections of other bundles such as F =Hom C ( T ˙Σ , E ) over this domain when there is no danger of confusion. The space W k,p (Σ ) ⊂ W k,p ( E )is defined in this case as the W k,p -closure of the space of smooth sections of E withcompact support in Σ \ ∂ Σ . For some background discussion on Sobolev spaces ofsections of vector bundles, see Appendix A. In Lecture 2 we proved that for 1 < p < ∞ , weak solutions of class L p loc to linearCauchy-Riemann type equations are always smooth. Here is a global version of thatresult. Proposition . Suppose < p < ∞ and k ∈ N . If η ∈ L p ( E ) weakly satisfies D η ∈ W k − ,p ( F ) , then η ∈ W k,p ( E ) . Proof.
By induction, it suffices to show that if η ∈ W k − ,p and D η ∈ W k − ,p then η ∈ W k,p . We already know that this is true locally, so the task is to bound the W k,p -norm of η on the cylindrical ends. Pick an asymptotic trivialization and write D on one of the ends Z ± ∼ = ˙ U z as ¯ ∂ + S ( s, t ). Let us assume for concreteness that thepuncture is a positive one, and now consider the W k,p -norm of η on ( N, N +1) × S ⊂ ˙ U z for N ∈ N . Choosing a smooth bump function β : R × S → [0 ,
1] supported in( N − , N + 2) × S with β = 1 on [ N, N + 1] × S , we can use the usual ellipticestimate to write k η k W k,p (( N,N +1) × S ) ≤ k βη k W k,p (( N − ,N +2) × S ) ≤ c k ¯ ∂ ( βη ) k W k − ,p (( N − ,N +2) × S ) ≤ c k η k W k − ,p (( N − ,N +2) × S ) + c k ¯ ∂η k W k − ,p (( N − ,N +2) × S ) = c k η k W k − ,p (( N − ,N +2) × S ) + c k D η − Sη k W k − ,p (( N − ,N +2) × S ) ≤ c ′ k η k W k − ,p (( N − ,N +2) × S ) + c ′ k D η k W k − ,p (( N − ,N +2) × S ) . An important detail here is that the constants in these estimates can be assumedindependent of N : indeed, one can use shifts of the same cutoff function for any N ,and the C k − -norm of S on [ N − , N + 2] × S is also bounded uniformly in N since S ( s, t ) converges asymptotically to some S ∞ ( t ). We can therefore take the sum ofthis estimate for all N ∈ N , producing k η k W k,p (˚ Z ) ≤ c k η k W k − ,p (˚ Z + ) + c k D η k W k − ,p (˚ Z + ) . (cid:3) Corollary . For < p < ∞ , any weak solution η ∈ L p ( E ) of D η = 0 issmooth, with derivatives of all orders decaying to at infinity. Proof.
Proposition 4.6 implies η ∈ W k,p ( E ) for every k ∈ N , so smoothnessfollows from the Sobolev embedding theorem. Moreover, suppose k and p are large ectures on Symplectic Field Theory enough to have a continuous inclusion W k,p ֒ → C m for some m ∈ N . Then thefiniteness of the W k,p -norm also implies that for each end ˙ U z = Z ± , k η k C m ( Z R ± ) ≤ c k η k W k,p (˚ Z R ± ) → R → ∞ . (cid:3) The local elliptic estimates for ¯ ∂ = ∂ s + J ∂ t in Lecture 2 applied to functionson ˚ D ⊂ C with compact support. Using a finite open covering with a subordinatepartition of unity, it is a straightforward matter to turn these local estimates intothe following global result (cf. [ Wend , Lemma 3.3.2]):
Proposition . If Σ ⊂ ˙Σ is a compact -dimensional submanifold withboundary, then there exists a constant c > such that k η k W k,p (Σ ) ≤ c k D η k W k − ,p (Σ ) + c k η k W k − ,p (Σ ) for all η ∈ W k,p (Σ ) . (cid:3) This unfortunately is unsufficient for the global problem under consideration,since one has to chop off the cylindrical ends of ˙Σ in order to obtain a compactdomain. We therefore supplement the previous local estimates with an asymptoticestimate.
Proposition . Suppose z ∈ Γ ± is a puncture such that the asymptotic op-erator A z is nondegenerate. Then on Z R ± ⊂ ˙ U z for sufficiently large R ≥ , thereexists a constant c > such that k η k W k,p (˚ Z R ± ) ≤ c k D η k W k − ,p (˚ Z R ± ) for all η ∈ W k,p ( ˚ Z R ± ) . Remark . Recall that W k,p ( ˚ Z R ± ) denotes the W k,p -closure of C ∞ ( ˚ Z R ± ), sosuch functions remain in W k,p if they are extended as zero to larger domains con-taining ˚ Z R ± . Note that functions of class W k,p on ˚ Z R ± need not actually have compactsupport; in fact C ∞ is dense in W k,p ( R × S ), see § A.4.The proof of this requires a basic result about translation-invariant Cauchy-Riemann type operators on the cylinder. Other than the elliptic estimates we dis-cussed in Lecture 2, this is the main analytical ingredient that makes all Floer-typetheories in symplectic geometry work.
Theorem . Suppose k ∈ N , < p < ∞ , and A = − J ∂ t − S ( t ) is anondegenerate asymptotic operator on the trivial Hermitian vector bundle S × R n → S . Then the operator ∂ s − A = ∂ s + J ∂ t + S ( t ) : W k,p ( R × S , R n ) → W k − ,p ( R × S , R n ) is an isomorphism. (cid:3) A detailed proof of this result for k = 1 can be found in [ Sal99 , Lemma 2.4],and the general result follows easily from this using regularity (Proposition 4.6). Iwill not attempt to reproduce the proof in Salamon’s notes here since it is somewhat Chris Wendl involved, but let us informally sketch the first step, which is the interesting part.The goal is to prove that D := ∂ s − A is an invertible operator from H ( R × S )to L ( R × S ). To gain some intuition on this, consider the special case where theasymptotic operator is of the form A = − i∂ t − C for some constant C ∈ R . Onecan then write down an inverse of D explicitly by combining a Fourier transformin the s variable with a Fourier series in the t variable. That is, sufficiently nicefunctions u on R × S can be expressed as u ( s, t ) = X k ∈ Z Z R ˆ u k ( σ ) e πiσs e πikt dσ, where the hybrid Fourier transform/series ˆ u depends on a continuous variable σ ∈ R and a discrete variable k ∈ Z . One can then obtain ˆ u from u byˆ u k ( σ ) = Z R × S u ( s, t ) e − πiσs e − πikt ds dt, and we have the usual derivative formulas c ∂ s u k ( σ ) = 2 πiσ ˆ u k ( σ ) and c ∂ t u k ( σ ) =2 πik ˆ u k ( σ ). The relation ( ∂ s + i∂ t + C ) u = f therefore produces an inversion formulaof the form ˆ u k ( σ ) = ˆ f k ( σ )2 πiσ − πk + C .
This is a nice formula and produces from any f ∈ L an element u ∈ H unless C ∈ π Z , in which case the denominator has a singularity. This condition means C must not be an eigenvalue of − i∂ t , or in other words, A = − i∂ t − C is nondegenerate.One can perhaps imagine carrying out a similar argument in the general case usingan orthonormal set of eigenfunctions for A in place of the functions e πikt ; this ispresumably part of the idea behind the actual proof in [ Sal99 ], which uses stronglycontinuous semigroups generated by the self-adjoint operator A . Proof of Proposition 4.9.
Write D = ∂ s + J ∂ t + S ( s, t ) and D = ∂ s + J ∂ t + S ∞ ( t ) in an asymptotic trivialization on ˙ U z = Z ± , where the nondegenerateasymptotic operator is A = − J ∂ t − S ∞ ( t ) and we assume k S − S ∞ k C k − ( Z R ± ) → R → ∞ . For η ∈ W k,p ( ˚ Z R ± ), there is a canonical extension η ∈ W k,p ( R × S ) that equals zerooutside Z R ± , so by Theorem 4.11 we have k η k W k,p (˚ Z R ± ) = k η k W k,p ( R × S ) ≤ c k D η k W k − ,p ( R × S ) = c k D η k W k − ,p ( R × S ) . Rewriting this in terms of D gives k η k W k,p (˚ Z R ± ) ≤ c k D η k W k − ,p ( Z R ± ) + c k ( S ∞ − S ) η k W k − ,p ( Z R ± ) , Recall from Lecture 3 that the spectrum σ ( A ) of an arbitrary asymptotic operator A alwaysconsists only of isolated real eigenvalues, thus one can find λ ∈ R for which λ − A : H ( S ) → L ( S ) is invertible. Its inverse, also known as the resolvent , then defines a compact self-adjointoperator ( λ − A ) − : L ( S ) → L ( S ) due to the compact inclusion H ( S ) ֒ → L ( S ). Thespectral theorem for compact self-adjoint operators now provides an orthonormal basis of L ( S )consisting of eigenfunctions of ( λ − A ) − , which are also eigenfunctions of A . ectures on Symplectic Field Theory where the constants c > R . For this reason, we are free to make R ≥ C k − -norm of S ∞ − S on Z R ± less than an arbitrarilysmall number δ >
0, in which case the above gives k η k W k,p (˚ Z R ± ) ≤ c k D η k W k − ,p ( Z R ± ) + cδ k η k W k − ,p ( Z R ± ) , and thus by the inclusion W k − ,p ֒ → W k,p , k η k W k,p (˚ Z R ± ) ≤ c − cδ k D η k W k − ,p ( Z R ± ) . (cid:3) The standard approach for proving that elliptic operators are Fredholm beginsby proving that they are semi-Fredholm , meaning dim ker D < ∞ and im D isclosed. In most settings, it is not hard to show that local elliptic estimates give riseto global estimates of the form k η k W k,p ≤ c k D η k W k − ,p + k η k W k − ,p . The step fromthese estimates to the semi-Fredholm property is then provided by the followinglemma. Lemma . Suppose X , Y and Z are Banach spaces, T ∈ L ( X, Y ) , K ∈ L ( X, Z ) is compact, and there is a constant c > such that for all x ∈ X , (4.3) k x k X ≤ c k T x k Y + c k K x k Z . Then ker T is finite dimensional and im T is closed. Proof.
A vector space is finite dimensional if and only if the unit ball in thatspace is a compact set, so we begin by proving the latter holds for ker T . Suppose x k ∈ ker T is a bounded sequence. Then since K is a compact operator, K x k hasa convergent subsequence in Z , which is therefore Cauchy. But (4.3) then impliesthat the corresponding subsequence of x k in X is also Cauchy, and thus converges.Since we now know ker T is finite dimensional, we also know there is a closedcomplement V ⊂ X with ker T ⊕ V = X . Then the restriction T | V has the sameimage as T , thus if y ∈ im T , there is a sequence x k ∈ V such that T x k → y .We claim that x k is bounded. If not, then T ( x k / k x k k X ) → K ( x k / k x k k X )has a convergent subsequence, so (4.3) implies that a subsequence of x k / k x k k X alsoconverges to some x ∞ ∈ V with k x ∞ k = 1 and T x ∞ = 0, a contradiction. But nowsince x k is bounded, K x k also has a convergent subsequence and T x k converges byassumption, thus (4.3) yields also a convergent subsequence of x k , whose limit x satisfies T x = y . This completes the proof that im T is closed. (cid:3) In the analysis of closed J -holomorphic curves, one makes use of the above lemmaby placing the inclusion W k − ,p ֒ → W k,p in the role of the compact operator K .Unfortunately, W k − ,p ֒ → W k,p is not compact when the domain ˙Σ has cylindricalends; in contrast to the case of a compact domain, there is no way to write the normon the ends as a finite sum of norms for functions on domains of finite measure. Tocircumvent this problem, let Σ R ⊂ ˙Σ Chris Wendl denote the compact complement of the ends ˚ Z R ± ⊂ ˙ U z for all z ∈ Γ. Lemma . Fix k ∈ N and < p < ∞ , and assume all the A z are non-degenerate. Then for sufficiently large R > , there exists a constant c > suchthat k η k W k,p ( ˙Σ) ≤ c k D η k W k − ,p ( ˙Σ) + c k η k W k − ,p (Σ R ) for all η ∈ W k,p ( E ) . Proof.
Fix a smooth cutoff function β ∈ C ∞ (Σ R ) such that β | Σ R − ≡
1, andwrite ˙ U R Γ ⊂ ˙Σfor the union of all the ends ˚ Z R ± ⊂ ˙ U z for z ∈ Γ + ∪ Γ − . Then we can write any η ∈ W k,p ( E ) as η = βη +(1 − β ) η so that βη ∈ W k,p (Σ R ) and (1 − β ) η ∈ W k,p ( ˙ U R − ).Choosing R large enough to make Proposition 4.9 valid, we can apply this togetherwith Proposition 4.8 to show k η k W k,p ( ˙Σ) ≤ k βη k W k,p (Σ R ) + k (1 − β ) η k W k,p ( ˙ U R − ) ≤ c k D ( βη ) k W k − ,p (Σ R ) + c k βη k W k − ,p (Σ R ) + k D [(1 − β ) η ] k W k − ,p ( ˙ U R − ) . After applying the Leipbniz rule and absorbing the norms of β and ¯ ∂β into theconstants, this produces the stated inequality since the term involving the W k − ,p -norm of η on the cylindrical ends includes ¯ ∂ (1 − β ), which vanishes outside of Σ R . (cid:3) Lemma 4.12 is now applicable since the operator W k,p ( ˙Σ) → W k − ,p (Σ R ) : η η | Σ R involves the compact inclusion W k,p (Σ R ) ֒ → W k − ,p (Σ R ) and is thus compact. Corollary . If all the A z are nondegenerate, then D : W k,p ( E ) → W k − ,p ( F ) is semi-Fredholm. (cid:3) In order to show that coker D is also finite dimensional, we will apply the abovearguments to the formal adjoint of D , an operator whose kernel is naturally isomor-phic to the cokernel of D . Let us choose Hermitian bundle metrics h , i E on E and h , i F on F , and fix an area form d vol on ˙Σ that takes the form d vol = ds ∧ dt on the cylindrical ends. The formal adjoint of D is then defined as the uniquefirst-order linear differential operator D ∗ : Γ( F ) → Γ( E )that satisfies the relation h λ, D η i L ( F ) = h D ∗ λ, η i L ( E ) for all η ∈ C ∞ ( E ) , λ ∈ C ∞ ( F ) , ectures on Symplectic Field Theory where we use the real-valued L -pairings h η, ξ i L ( E ) := Re Z ˙Σ h η, ξ i E d vol , for η, ξ ∈ Γ( E ) , h α, λ i L ( F ) := Re Z ˙Σ h α, λ i F d vol , for α, λ ∈ Γ( F ) . The word “formal” refers to the fact that we are not viewing D ∗ as the adjoint ofan unbounded operator on a Hilbert space (cf. [ RS80 ]); that would be a strongercondition.
Exercise . Show that D ∗ is well defined and, for suitable choices of complexlocal trivializations of E and F and holomorphic coordinates on open subsets U ⊂ ˙Σ,can be written locally as D ∗ = − ∂ + A : C ∞ ( U , R n ) → C ∞ ( U , R n )for some A ∈ C ∞ ( U , End( R n )), where ∂ := ∂ s − J ∂ t .The formula in the above exercise reveals that D ∗ is also an elliptic operator and thus has the same local properties as D ; indeed, − ∂ + A can be transformedinto ¯ ∂ + B for some zeroth-order term B if we conjugate it by a suitable complex-antilinear change of trivialization. In particular, our local estimates for D and theirconsequences, notably Proposition 4.8, are all equally valid for D ∗ .To obtain suitable asymptotic estimates for D ∗ , let us fix asymptotic trivializa-tions τ of E , use the corresponding trivializations of F over the ends as describedin § compatiblewith the asymptotically Hermitian structure of E whenever they are chosenin this way outside of a compact subset of ˙Σ. We can then express D as ¯ ∂ + S ( s, t )on ˙ U z = Z ± , and integrate by parts to obtain D ∗ = − ∂ + S ( s, t ) T . To identify this expression with a Cauchy-Riemann type operator, let C := (cid:18) − (cid:19) denote the R -linear transformation on R n = C n representing complex conjugation.Then since C anticommutes with J , we have( C − D ∗ C ) η = − C∂ s ( Cη ) + CJ ∂ t ( Cη ) + CS ( s, t ) T Cη = − ∂ s η − J ∂ t η + CS ( s, t ) T Cη = − ( ¯ ∂η − CS ( s, t ) T Cη )=: − ( ¯ ∂ + ¯ S ( s, t )) η, where we’ve defined ¯ S ( s, t ) := − CS ( s, t ) T C . Now if the asymptotic operator A z at z ∈ Γ ± is written in the chosen trivialization as A := − J ∂ s − S ∞ ( t ), the asymptotic Technically, this property of the formal adjoint is part of the definition of ellipticity: we call adifferential operator elliptic whenever (1) it has the properties necessary for proving fundamentalestimates using Fourier transforms as we did with ¯ ∂ in § Chris Wendl convergence of S ( s, t ) implies that similarly k ¯ S − ¯ S ∞ k C k ( Z R ± ) → R → ∞ for all k ∈ N , where ¯ S ∞ ( t ) := − CS ∞ ( t ) C. This defines a trivialized asymptotic operator A = − J ∂ t − ¯ S ∞ ( t ) to which − D ∗ is(after a suitable change of trivialization) asymptotic at the puncture z ; in particular,our proof of the global regularity result, Proposition 4.6, now also works for D ∗ .Finally, notice that A and − A are conjugate: indeed,( C − A C ) η = − CJ ∂ t ( Cη ) + CCS ∞ ( t ) C ( Cη ) = J ∂ t η + S ∞ ( t ) η = − A η. This implies that A is nondegenerate if and only if A is; applying this assumptionfor all of the A z , the proofs of Proposition 4.9 and Lemma 4.13 now also go throughfor D ∗ .We’ve proved: Proposition . Suppose D ∗ is defined with respect Hermitian bundle metricson E and F = Hom C ( T ˙Σ , E ) that are compatible with the asymptotically Hermitianstructure of E . If additionally all the asymptotic operators A z are nondegenerate,then D ∗ : W k,p ( F ) → W k − ,p ( E ) is semi-Fredholm, and its kernel is a space of smooth sections contained in W m,q ( F ) for all m ∈ N and q ∈ (1 , ∞ ) . (cid:3) Since ker D ∗ is now known to be finite dimensional, the next result completesthe proof of the Fredholm property for D by showing that its image has finitecodimension: Lemma . Under the same assumptions as in Proposition 4.16, W k − ,p ( F ) = im D + ker D ∗ . Proof.
Consider first the case k = 1. Since D : W ,p ( E ) → L p ( F ) is semi-Fredholm, its image is closed, hence im D + ker D ∗ is a closed subspace of L p ( F ).Then if im D + ker D ∗ = L p ( F ), the Hahn-Banach theorem provides a nontrivialelement α ∈ ( L p ( F )) ∗ ∼ = L q ( F ) for p + q = 1 such that(4.4) h D η + λ, α i L ( F ) = 0 for all η ∈ W ,p ( E ) , λ ∈ ker D ∗ . Choosing λ = 0, this implies in particular h D η, α i L ( F ) = 0 for all η ∈ C ∞ ( E ) , which means that α is a weak solution of class L q to the formal adjoint equation D ∗ α = 0. By Proposiiton 4.6, α is therefore smooth and belongs to ker D ∗ . Butthis contradicts (4.4) if we plug in η = 0 and λ = α , so this completes the proof for k = 1. In the case p = 2, one can forego the Hahn-Banach theorem and simply take an L -orthogonalcomplement. ectures on Symplectic Field Theory For k ≥
2, suppose α ∈ W k − ,p ( F ) ⊂ L p ( F ) is given: then the case k = 1provides elements η ∈ W ,p ( E ) and λ ∈ ker D ∗ such that D η + λ = α . SinceProposition 4.6 implies λ ∈ W m,q ( F ) for all m ∈ N and q ∈ (1 , ∞ ), we have D η = α − λ ∈ W k − ,p ( F ) and thus, by Prop. 4.6 again, η ∈ W k,p ( E ), completingthe proof for all k ∈ N . (cid:3) The proof of Theorem 4.3 is now complete, but as long as we’re talking aboutthe formal adjoint, let us take note of a few more properties that will be useful inthe future. Assume from now on that all the assumptions of Proposition 4.16 aresatisfied. We can now strengthen Lemma 4.17 as follows.
Proposition . W k − ,p ( F ) = im D ⊕ ker D ∗ and W k − ,p ( E ) = im D ∗ ⊕ ker D .In particular, the projections defined by these splittings give isomorphisms coker D ∼ = ker D ∗ and coker D ∗ ∼ = ker D , thus D ∗ : W k,p ( F ) → W k − ,p ( E ) is a Fredholm operator with ind D ∗ = − ind D . Proof.
By Lemma 4.17, the first splitting follows if we can show that im D ∩ ker D ∗ = { } . Recall first (see § A.4) that C ∞ ( ˙Σ) is dense in W k,p ( ˙Σ) for every k ≥ p ∈ [1 , ∞ ), so the definition of the formal adjoint implies via density andH¨older’s inequality that if 1 < p, q < ∞ and p + q = 1,(4.5) h λ, D η i L ( F ) = h D ∗ λ, η i L ( E ) for all η ∈ W ,p ( E ) , λ ∈ W ,q ( F ) . Now suppose λ ∈ im D ∩ ker D ∗ and write λ = D η , assuming η ∈ W k,p ( E ). Reg-ularity implies that since D ∗ λ = 0, λ ∈ W ,q ( F ), where q can be chosen to satisfy p + q = 1. We can therefore apply (4.5) and obtain h λ, λ i L ( F ) = h λ, D η i L ( F ) = h D ∗ λ, η i L ( E ) = 0 , hence λ = 0.The proof that W k − ,p ( E ) = im D ∗ ⊕ ker D is analogous. (cid:3) This result hints at the fact that D ∗ is in fact—under some natural extraassumptions—globally equivalent to another Cauchy-Riemann type operator. Tosee this, let us impose a further constraint on the relation between the Hermitianbundle metrics h , i E and h , i F . Note that since the area form d vol is necessarily j -invariant, it induces a Hermitian structure on T ˙Σ, namely h X, Y i Σ := d vol( X, jY ) + i d vol(
X, Y ) , which matches the standard bundle metric in the trivializations over the ends definedvia the cylindrical coordinates. This induces real-linear isomorphisms from T ˙Σ tothe complex-linear and -antilinear parts of the complexified cotangent bundle, T ˙Σ → Λ , T ∗ ˙Σ : X X , := h X, ·i Σ ,T ˙Σ → Λ , T ∗ ˙Σ : X X , := h· , X i Σ , where the first isomorphism is complex antilinear and the second is complex linear.We use these to define Hermitian bundle metrics on Λ , T ∗ ˙Σ and Λ , T ∗ ˙Σ in terms Chris Wendl of the metric on T ˙Σ; note that this is a straightforward definition for Λ , T ∗ ˙Σ, butsince the isomorphism to Λ , T ∗ ˙Σ is complex antilinear , we really mean h X , , Y , i Σ := h Y, X i Σ for X, Y ∈ T ˙Σ . Now observe that as a vector bundle with complex structure λ J ◦ λ , F =Hom C ( T ˙Σ , E ) is naturally isomorphic to the complex tensor product F = Λ , T ∗ Σ ⊗ E. We can therefore make a natural choice for h , i F as the tensor product metricdetermined by h , i Σ and h , i E . It is easy to check that this choice is compatiblewith the asymptotically Hermitian structure of E .Next, we notice that the area form d vol also induces a natural complex bundleisomorphism E → Hom C ( T ˙Σ , F ) . Indeed, the right hand side is canonically isomorphic to the complex tensor productHom C ( T ˙Σ , F ) = Λ , T ∗ ˙Σ ⊗ F = Λ , T ∗ ˙Σ ⊗ Λ , T ∗ ˙Σ ⊗ E, and Λ , T ∗ ˙Σ ⊗ Λ , T ∗ ˙Σ is isomorphic to the trivial complex line bundle ǫ := ˙Σ × C → ˙Σ via Λ , T ∗ ˙Σ ⊗ Λ , T ∗ ˙Σ → ǫ : X , ⊗ Y , X , ( Y ) = h X, Y i Σ . Exercise . Assuming h , i F is chosen as the tensor product metric describedabove, show that under the natural identification of E with Hom C ( T ˙Σ , F ), − D ∗ : Γ( F ) → Ω , ( ˙Σ , F )satisfies the Leibniz rule − D ∗ ( f λ ) = ( ∂f ) λ + f ( − D ∗ λ )for all f ∈ C ∞ ( ˙Σ , R ), where ∂f ∈ Ω , ( ˙Σ) denotes the complex-valued (1 , df − i df ◦ j .We might summarize this exercise by saying that − D ∗ is an “anti-Cauchy-Riemann type” operator on F . But such an object is easily transformed into anhonest Cauchy-Riemann type operator: let ¯ F denote the conjugate bundle to F ,which we define as the same real vector bundle F but with the sign of its complexstructure reversed, so λ
7→ − J ◦ λ . Now there is a canonical isomorphismHom C ( T ˙Σ , F ) = Hom C ( T ˙Σ , ¯ F ) , and the same operator defines a real-linear map − D ∗ : Γ( ¯ F ) → Ω , ( ˙Σ , ¯ F )which satisfies our usual Leibniz rule for Cauchy-Riemann type operators.Its asymptotic behavior also fits into the scheme we’ve been describing: wehave already seen this by computing D ∗ on the ends with respect to asymptotictrivializations. To express this in trivialization-invariant language, observe that eachof the Hermitian bundles ( E z , J z , ω z ) over S for z ∈ Γ has a conjugate bundle ¯ E z with complex structure − J z and symplectic structure − ω z ; its natural Hermitian ectures on Symplectic Field Theory inner product is then the complex conjugate of the one on E z . The asymptoticoperator A z on E z can be expressed as − J z b ∇ t , where b ∇ t is a symplectic connectionon ( E z , ω z ). Then b ∇ t is also a symplectic connection on ( ¯ E z , − ω z ), so we naturallyobtain an asymptotic operator on ¯ E z in the form(4.6) A z := − A z : Γ( ¯ E z ) → Γ( ¯ E z ) , where the sign reversal arises from the reversal of the complex structure. One cancheck that if we choose a unitary trivialization of E z and the conjugate trivializationof ¯ E z , this relationship between A z and A z produces precisely the relationshipbetween A = − J ∂ t − S ∞ ( t ) and A = − J ∂ t − ¯ S ∞ ( t ) that we saw previously, with¯ S ∞ ( t ) = − CS ∞ ( t ) C . Let us summarize all this with a theorem. Theorem . Assume h , i F is chosen to be the tensor product metric on F = Λ , T ∗ Σ ⊗ E induced by h , i E and the area form d vol . Then under theisomorphism induced by d vol from E to Hom C ( T ˙Σ , F ) and the natural identificationof the latter with its conjugate Hom C ( T ˙Σ , ¯ F ) , the operator − D ∗ : Γ( F ) → Γ( E ) defines a linear Cauchy-Riemann type operator on the conjugate bundle ¯ F , − D ∗ : Γ( ¯ F ) → Ω , ( ˙Σ , ¯ F ) , and it is asymptotic at each puncture z ∈ Γ to the conjugate asymptotic operator (4.6) . (cid:3) ECTURE 5
The index formulaContents
As in the previous lecture, let D denote a linear Cauchy-Riemann type operatoron an asymptotically Hermitian vector bundle E of complex rank m over a puncturedRiemann surface ( ˙Σ = Σ \ (Γ + ∪ Γ − ) , j ), and assume that D is asymptotic at eachpuncture z ∈ Γ to a nondegenerate asymptotic operator A z on the asymptoticbundle ( E z , J z , ω z ) over S . Writing F := Hom C ( T ˙Σ , E )for the bundle of complex-antilinear homomorphisms T ˙Σ → E , the main result ofthe previous lecture was that D : W k,p ( E ) → W k − ,p ( F )is Fredholm for any k ∈ N and p ∈ (1 , ∞ ), and its kernel and index do not dependon k or p . The main goal of this lecture is to compute ind( D ) ∈ Z .The index will depend on the Conley-Zehnder indices µ τ CZ ( A z ) ∈ Z introducedin Lecture 3, but since these depend on arbitrary choices of unitary trivializations τ ,we need a way of selecting preferred trivializations. The most natural condition is torequire that every ( E z , J z , ω z ) be endowed with a unitary trivialization such that thecorresponding asymptotic trivializations of ( E, J ) extend to a global trivialization ;if there is only one puncture z , for instance, then this condition determines µ τ CZ ( A z )uniquely. This convention has been used to state the formula for ind( D ) in severalof the standard references, e.g. in [ HWZ99 ]. We would prefer however to state aformula which is also valid when Γ = ∅ and E → Σ is nontrivial. One way to do Note that (
E, J ) is always globally trivializable unless Γ = ∅ , as a punctured surface can beretracted to its 1-skeleton. Chris Wendl this is by allowing completely arbitrary asymptotic trivializations, but introducinga topological invariant to measure their failure to extend globally over E . Definition . Fix a compact oriented surface S with boundary. The relativefirst Chern number associates to every complex vector bundle ( E, J ) over S andtrivialization τ of E | ∂S an integer c τ ( E ) ∈ Z satisfying the following properties:(1) If ( E, J ) → S is a line bundle, then c τ ( E ) is the signed count of zeroes fora generic smooth section η ∈ Γ( E ) that appears as a nonzero constant at ∂S with respect to τ .(2) For any two bundles ( E , J ) and ( E , J ) with trivializations τ and τ respectively over ∂S , c τ ⊕ τ ( E ⊕ E ) = c τ ( E ) + c τ ( E ) . These two conditions uniquely determine c τ ( E ) for all complex vector bundlessince bundles of higher rank can always be split into direct sums of line bundles.The definition clearly matches the usual first Chern number c ( E ) when ∂S = ∅ ,and it extends in an obvious way to the category of asymptotically Hermitian vectorbundles with asymptotic trivializations. Exercise . Given two distinct choices of asymptotic trivializations τ and τ for an asymptotically Hermitian bundle E of rank m , show that c τ ( E ) = c τ ( E ) − deg( τ ◦ τ − ) , where deg( τ ◦ τ − ) ∈ Z denotes the sum over all punctures of the winding numbersof the determinants of the transition maps S → U( m ). Exercise . Combining Exercise 5.2 above with Exercise 3.37, show that forour asymptotically Hermitian vector bundle E with Cauchy-Riemann type operator D and asymptotic operators A z , the number2 c τ ( E ) + X z ∈ Γ + µ τ CZ ( A z ) − X z ∈ Γ − µ τ CZ ( A z )is independent of the choice of asymptotic trivializations τ .The above exercise shows that the right hand side of the following index formulais independent of all choices. Theorem . The Fredholm index of D is given by ind D = mχ ( ˙Σ) + 2 c τ ( E ) + X z ∈ Γ + µ τ CZ ( A z ) − X z ∈ Γ − µ τ CZ ( A z ) , where m = rank C E and τ is an arbitrary choice of asymptotic trivializations. Caution: to compute this winding number at a negative puncture using cylindrical coordinates( s, t ) ∈ ( −∞ , × S , one must traverse {− s } × S for s ≫ wrong direction , as this isconsistent with the orientation induced on {− s } × S as a boundary component of a large compactsubdomain of ˙Σ. ectures on Symplectic Field Theory Notation.
Throughout this lecture, we shall denote the integer on the righthand side in Theorem 5.4 by I ( D ) := mχ ( ˙Σ) + 2 c τ ( E ) + X z ∈ Γ + µ τ CZ ( A z ) − X z ∈ Γ − µ τ CZ ( A z ) ∈ Z . Our goal is thus to prove that ind( D ) = I ( D ).When Γ = ∅ , Theorem 5.4 is equivalent to the classical Riemann-Roch formula,which is more often stated for holomorphic vector bundles over a closed Riemannsurface (Σ , j ) with genus g as(5.1) ind C ( D ) = m (1 − g ) + c ( E ) . This formula assumes that the Cauchy-Riemann type operator D is complex linear,but an arbitrary real-linear Cauchy-Riemann operator is then of the form D = D + B , where the zeroth-order term B ∈ Γ(Hom R ( E, F )) defines a compact perturbationsince the inclusion W k,p (Σ) ֒ → W k − ,p (Σ) is compact. It follows that D has the same real Fredholm index as D , namely twice the complex index shown on the right handside of (5.1), which matches what we see in Theorem 5.4. Remark . Now seems a good moment to clarify explicitly that all dimensions(and therefore also Fredholm indices) in this lecture are real dimensions, not complexdimensions, unless otherwise stated.Reduction to the complex-linear case does not work in general if there are punc-tures: it remains true that arbitrary Cauchy-Riemann type operators can be writtenas D = D + B where D is complex linear, but the perturbation introduced bythe zeroth-order term B is not compact since W k,p ( ˙Σ) ֒ → W k − ,p ( ˙Σ) is not compactwhen Γ = ∅ . Another indication that this idea cannot work is the fact that whilethe formula in Theorem 5.4 always gives an even integer when Γ = ∅ , it can be oddwhen there are punctures, in which case D clearly cannot have the same index isany complex-linear operator. Our proof will therefore have to deal with more thanjust the complex category.The punctured version of Theorem 5.4 was first proved by Schwarz in his the-sis [ Sch95 ], its main purpose at the time being to help define algebraic operations(notably the pair-of-pants product ) in Hamiltonian Floer homology. Schwarz’s proofused a “linear gluing” construction that gives a relation between indices of opera-tors on bundles over surfaces obtained by gluing together constituent surfaces alongmatching cylindrical ends. Since any surface with ends can be “capped off” to forma closed surface, one obtains the general index formula if one already knows how tocompute it for closed surfaces and for planes (i.e. caps). For the latter, it is simpleenough to write down model Cauchy-Riemann operators on planes and computetheir kernels and cokernels explicitly, so in this way the general case is reduced tothe classical Riemann-Roch formula. An analogous linear gluing argument for com-pact surfaces with boundary is used in [
MS04 , Appendix C] to reduce the generalRiemann-Roch formula to an explicit computation for Cauchy-Riemann operatorson the disk with a totally real boundary condition. Chris Wendl
In this lecture, we will follow a different path and use an argument that wasfirst sketched by Taubes for the closed case in [
Tau96a , § Ger ]. The argument is(in my opinion) analytically somewhat easier than the more standard approaches,and in addition to proving the formula we need for punctured surfaces, it produces anew proof in the closed case without assuming the classical Riemann-Roch formula.It also provides a gentle preview of two analytical phenomena that will later assumeprominent roles in our discussion of SFT: bubbling and gluing .To see the idea behind Taubes’s argument, we can start by noticing an apparentnumerical coincidence in the closed case. Assume (
E, J ) is a complex line bundle overa closed Riemann surface (Σ , j ), and D : Γ( E ) → Γ( F ) = Ω , (Σ , E ) is a Cauchy-Riemann type operator. We know that ind( D ) = ind( D + B ) for any zeroth-orderterm B ∈ Γ(Hom R ( E, F )). But E and F are both complex vector bundles, so B can always be split uniquely into its complex-linear and complex-antilinear parts,i.e. there is a natural splitting of Hom R ( E, F ) into a direct sum of complex linebundles Hom R ( E, F ) = Hom C ( E, F ) ⊕ Hom C ( E, F ) . Out of curiosity, let’s compute the first Chern number of the second factor; this willbe the signed count of zeroes of a generic complex- antilinear zeroth-order perturba-tion. To start with, note thatHom C ( E, F ) = Hom C ( E, C ) ⊗ F, and then observe that Hom C ( E, C ) and E are isomorphic: indeed, any Hermitianbundle metric h , i E on E gives rise to a bundle isomorphism E → Hom C ( E, C ) : η
7→ h· , η i E . We thus have Hom C ( E, F ) ∼ = E ⊗ F , so c (Hom C ( E, F )) = c ( E ) + c ( F ). We cancompute c ( F ) by the same trick since F = Hom C ( T Σ , E ) = Hom C ( T Σ , C ) ⊗ E ∼ = T Σ ⊗ E, so c ( F ) = c ( T Σ) + c ( E ) = χ (Σ) + c ( E ), and thus c (Hom C ( E, F )) = χ (Σ) + 2 c ( E ) . Since we’re looking at a line bundle over a surface without punctures, this numberis the same as I ( D ). This coincidence is too improbable to ignore, and indeed, itturns out not to be coincidental. Here is an informal statement of a result that wewill later prove a more precise version of in order to deduce Theorem 5.4. “Theorem”. Given a Cauchy-Riemann type operator D : H ( E ) → L ( F ) on aline bundle ( E, J ) over a closed Riemann surface (Σ , j ) , choose a complex-antilinearzeroth-order perturbation B ∈ Γ(Hom C ( E, F )) whose zeroes are all nondegenerate. Here the complex structure on Hom R ( E, F ) and its subbundles is defined in terms of thecomplex structure of F , i.e. it sends B ∈ Hom R ( E, F ) to J ◦ B ∈ Hom R ( E, F ). We are assuming as usual that Hermitian inner products are complex antilinear in the firstargument and linear in the second. ectures on Symplectic Field Theory Then for sufficiently large σ > , ker( D + σB ) is approximately spanned by -dimensional spaces of sections with support localized near the positive zeroes of B .In particular, dim ker( D + σB ) equals the number of positive zeroes of B . To deduce ind( D ) = I ( D ) from this, we need to apply the same trick to theformal adjoint D ∗ . As we will review in § − D ∗ can be regarded under certainnatural assumptions as a Cauchy-Riemann type operator on the bundle ¯ F conjugateto F , and the formal adjoint of D + σB then gives rise to a Cauchy-Riemann typeoperator of the form − D ∗ + σB ′ : Γ( ¯ F ) → Γ( ¯ E ) = Ω , (Σ , ¯ F ) , where B ′ : ¯ F → ¯ E is also complex antilinear and has the same zeroes as B , but withopposite signs. Applying the above “theorem” to − D ∗ thus identifies ker( D + σB ) ∗ for sufficiently large σ > negative zeroes of B . This givesind( D ) = ind( D + σB ) = dim ker( D + σB ) − dim ker( D + σB ) ∗ = c (Hom C ( E, F )) = I ( D ) . It’s worth mentioning that the “large perturbation” argument we’ve just sketchedis only one simple example of an idea with a long and illustrious history: anothersimple example is the observation by Witten [
Wit82 ] that after choosing a Morsefunction on a Riemannian manifold, certain large deformations of the de Rhamcomplex lead to an approximation of the Morse complex, with generators of the deRham complex having support concentrated near the critical points of the Morsefunction—this yields a somewhat novel proof of de Rham’s theorem. A much deeperexample is Taubes’s isomorphism [
Tau96b ] between the Seiberg-Witten invariantsof symplectic 4-manifolds and certain holomorphic curve invariants: here also, theidea is to consider a large compact perturbation of the Seiberg-Witten equations andshow that, in the limit where the perturbation becomes infinitely large, solutions ofthe Seiberg-Witten equations localize near J -holomorphic curves. For a more recentexploration of this idea in the context of Dirac operators, see [ Mar ].Before proceeding with the details, let us fix two simplifying assumptions thatcan be imposed without loss of generality:
Assumption . ( E, J ) has complex rank . Indeed, an asymptotically Hermitian bundle E of complex rank m ∈ N alwaysadmits a decomposition into asymptotically Hermitian line bundles E = E ⊕ . . . ⊕ E m , producing a corresponding splitting of the target bundle F = F ⊕ . . . ⊕ F m .The operator D need not respect these splittings, but it is always homotopic throughFredholm operators to one that does: we saw in Theorem 3.34 that the asymptoticoperators A z are homotopic through nondegenerate asymptotic operators to anyother operators A ′ z that have the same Conley-Zehnder indices, so one can choose A ′ z to respect the splitting. Any homotopy of Cauchy-Riemann operators followingsuch a homotopy of nondegenerate asymptotic operators then produces a continuousfamily of Fredholm operators by the main result of Lecture 4, implying that theirindices do not change. The general index formula then follows from the line bundle Chris Wendl case since any two Cauchy-Riemann type Fredholm operators D and D over thesame Riemann surface satisfyind( D ⊕ D ) = ind( D ) + ind( D ) and I ( D ⊕ D ) = I ( D ) + I ( D ) . Assumption . k = 1 and p = 2 . This means we will concretely be considering the operator D : H ( E ) → L ( F ) , where H as usual is an abbreviation for W , . This assumption is clearly harmlesssince we know that ind D does not depend on the choice of k and p . For the beginning of this section we can drop the assumption that (
E, J ) is aline bundle and assume rank C E = m ∈ N , though later we will again set m = 1.Recall from the end of Lecture 4 that if we fix global Hermitian structures h , i E and h , i F on ( E, J ) and (
F, J ) respectively and an area form d vol on ˙Σ that matches ds ∧ dt on the cylindrical ends, then D has a formal adjoint D ∗ : Γ( F ) → Γ( E )satisfying h λ, D η i L ( F ) = h D ∗ λ, η i L ( E ) for all η ∈ H ( E ) , λ ∈ H ( F ) . Here the real-valued L pairings are defined by h η, ξ i L ( E ) := Re Z ˙Σ h η, ξ i E d vol for η, ξ ∈ Γ( E ) , and similarly for sections of F . The essential features of the formal adjoint arethat ker D ∗ ∼ = coker D and coker D ∗ ∼ = ker D , hence ind( D ∗ ) = − ind( D ). Recallmoreover that d vol induces a natural Hermitian bundle metric on ˙Σ by h· , ·i Σ = d vol( · , j · ) + i d vol( · , · ) , which determines a bundle isomorphism T ˙Σ → Λ , T ∗ ˙Σ : X X , := h· , X i Σ , as well as a complex- antilinear isomorphism T ˙Σ → Λ , T ∗ ˙Σ : X X , := h X, ·i Σ . If h , i F is then chosen to be the tensor product metric determined via the naturalisomorphism F = Hom C ( T ˙Σ , E ) = Λ , T ∗ ˙Σ ⊗ E = T ˙Σ ⊗ E, then E admits a natural isomorphism to Λ , T ∗ ˙Σ ⊗ F such that − D ∗ : Γ( F ) → Γ( E ) = Ω , ( ˙Σ , F )becomes an anti-Cauchy-Riemann type operator, i.e. it satisfies the Leibniz rule − D ∗ ( f λ ) = ( ∂f ) λ + f ( − D ∗ λ ) ectures on Symplectic Field Theory for all f ∈ C ∞ ( ˙Σ , R ), with ∂f := df − i df ◦ j ∈ Ω , ( ˙Σ). Equivalently, − D ∗ definesa Cauchy-Riemann type operator on the conjugate bundle ¯ F → ˙Σ, defined as thereal bundle F → ˙Σ but with the sign of its complex structure reversed; we shalldistinguish this Cauchy-Riemann operator from − D ∗ by writing it as − D ∗ : Γ( ¯ F ) → Ω , ( ˙Σ , ¯ F ) , though it is technically the same operator. Recall that the identity map defines anatural complex-antilinear isomorphism between any complex vector bundle and itsconjugate bundle; we shall denote this isomorphism generally by E → ¯ E : v ¯ v, so in particular it satisfies cv = ¯ c ¯ v for all scalars c ∈ C , and similarly D ∗ ¯ λ = D ∗ λ for λ ∈ Γ( F ). The asymptotic operators for − D ∗ are A z = − A z : Γ( ¯ E z ) → Γ( ¯ E z ) . Lemma . If τ is a choice of asymptotic trivialization on E and ¯ τ denotes the conjugate asymptotic trivialization , then c ¯ τ ( ¯ E ) = − c τ ( E ) , and µ ¯ τ CZ ( A z ) = − µ τ CZ ( A z ) for all z ∈ Γ . Proof.
Assuming E is a line bundle, suppose η is a generic section of E thatmatches a nonzero constant with respect to τ on the cylindrical ends, so c τ ( E ) isthe signed count of zeroes of η . Then ¯ η ∈ Γ( ¯ E ) is similarly a nonzero constant onthe ends with respect to ¯ τ , but the signs of its zeroes are opposite those of η becausethey are defined as winding numbers with respect to conjugate local trivializations.This proves c ¯ τ ( ¯ E ) = − c τ ( E ).The Conley-Zehnder indices can be computed from the formula µ τ CZ ( A z ) = α τ + ( A z ) + α τ − ( A z ) , see Theorem 3.36. Here α τ − ( A z ) is the largest possible winding number relative to τ of an eigenfunction for A z with negative eigenvalue, and α τ + ( A z ) is the smallestpossible winding number with positive eigenvalue. The eigenfunctions of A z = − A z are the same, but the signs of their eigenvalues are reversed, and the signs of theirwinding numbers are also reversed because they must be measured relative to theconjugate trivialization, thus α ¯ τ ± ( A z ) = − α τ ∓ ( A z ) , implying µ ¯ τ CZ ( A z ) = α ¯ τ + ( A z ) + α ¯ τ − ( A z ) = − α τ − ( A z ) − α τ + ( A z ) = − µ τ CZ ( A z ) . The above calculations are all valid for line bundles, but the general case followsby taking direct sums. (cid:3) If τ : E | U → U × C m is a local trivialization of E with τ ( v ) = ( z, w ), the conjugate trivialization¯ τ : ¯ E | U → U × C m is defined by ¯ τ (¯ v ) = ( z, ¯ w ). Chris Wendl
We are now able to show that Theorem 5.4 is consistent with what we alreadyknow about the formal adjoint.
Proposition . I ( − D ∗ ) = − I ( D ) . Proof.
Under the isomorphism F = Λ , T ∗ ˙Σ ⊗ E = T ˙Σ ⊗ E , an asymptotictrivialization τ on E induces an asymptotic trivialization ∂ s ⊗ τ on F , where ∂ s denotes the asymptotic trivialization of T ˙Σ defined via an outward pointing vectorfield on the cylindrical ends. Counting zeroes of vector fields then proves c ∂ s ( T ˙Σ) = χ ( ˙Σ), so c ∂ s ⊗ τ ( F ) = c ∂ s ⊗ τ ( T ˙Σ ⊗ E ) = mc ∂ s ( T ˙Σ) + c τ ( E ) = mχ ( ˙Σ) + c τ ( E ) . Applying Lemma 5.8 to the conjugate bundle then gives c ∂ s ⊗ τ ( ¯ F ) = − mχ ( ˙Σ) − c τ ( E ) . The unitary trivializations of the asymptotic bundles ¯ E z corresponding to ∂ s ⊗ τ are simply ¯ τ , thus using Lemma 5.8 again for the Conley-Zehnder terms, I ( − D ∗ ) = mχ ( ˙Σ) + 2 c ∂ s ⊗ τ ( ¯ F ) + X z ∈ Γ + µ ¯ τ CZ ( A z ) − X z ∈ Γ − µ ¯ τ CZ ( A z )= − mχ ( ˙Σ) − c τ ( E ) − X z ∈ Γ + µ τ CZ ( A z ) + X z ∈ Γ − µ τ CZ ( A z )= − I ( D ) . (cid:3) We next consider the effect of an antilinear zeroth-order perturbation on theformal adjoint. By “antilinear zeroth-order perturbation,” we generally mean asmooth section B ∈ Γ(Hom C ( E, F )) . It is perhaps easier to understand B in terms of the conjugate bundle ¯ E : indeed,there exists a unique β ∈ Γ(Hom C ( ¯ E, F ))such that Bη = β ¯ η, and this correspondence defines a bundle isomorphism Hom C ( E, F ) = Hom C ( ¯ E, F ). Exercise . Assume X and Y are complex vector bundles over the samebase.(a) Show that ¯ X ⊗ ¯ Y is canonically isomorphic to the conjugate bundle of X ⊗ Y .(b) Show that Hom C ( ¯ X, ¯ Y ) is canonically isomorphic to the conjugate bundle ofHom C ( X, Y ), and Hom C ( ¯ X, ¯ Y ) is canonically isomorphic to the conjugatebundle of Hom C ( X, Y ).(c) Show that Λ , X := Hom C ( X, C ) is canonically isomorphic to the conjugatebundle of Λ , X := Hom C ( X, C ). ectures on Symplectic Field Theory Define the Cauchy-Riemann type operator D B := D + B : Γ( E ) → Γ( F ) = Ω , ( ˙Σ , E ) , so D B η = D η + β ¯ η . To write down D ∗ B , observe that since β : ¯ E → F is a complex-linear bundle map between Hermitian bundles, it has a complex-linear adjoint β † : F → ¯ E such that h β † λ, ¯ η i ¯ E = h λ, β ¯ η i F for λ ∈ F , ¯ η ∈ ¯ E. Here the bundle metric on ¯ E is defined by h ¯ η, ¯ ξ i ¯ E := h ξ, η i E . We then haveRe h λ, Bη i F = Re h λ, β ¯ η i F = Re h β † λ, ¯ η i ¯ E = Re h η, β † λ i E = Re h β † λ, η i E = Re h β † ¯ λ, η i E , where β † ∈ Γ(Hom C ( ¯ F , E )) denotes the image of β † ∈ Γ(Hom C ( F, ¯ E )) under thecomplex-antilinear identity map from Hom C ( F, ¯ E ) to its conjugate bundle (see Ex-ercise 5.10). The formal adjoint of D B is thus D ∗ B = D ∗ + B ∗ : Γ( F ) → Γ( E ) , where B ∗ : F → E is defined by B ∗ λ := β † ¯ λ. To write down the resulting Cauchy-Riemann type operator on ¯ F , we replace B ∗ : F → E with B ∗ : ¯ F → ¯ E , defined by B ∗ ¯ λ := B ∗ λ = β † λ, giving a Cauchy-Riemann operator − D ∗ B = − D ∗ + ( − B ∗ ) : Γ( ¯ F ) → Γ( ¯ E ) = Ω , ( ˙Σ , ¯ F ) . The point of writing down this formula is to make the following observations:
Lemma . The zeroth-order perturbation − B ∗ : ¯ F → ¯ E appearing in − D ∗ B has the following properties:(1) − B ∗ : ¯ F → ¯ E is complex antilinear;(2) There is a natural complex bundle isomorphism Hom C ( ¯ F , ¯ E ) = Hom C ( F, ¯ E ) that identifies − B ∗ with − β † ;(3) If m = 1 and B ∈ Γ(Hom C ( E, F )) has only nondegenerate zeroes, then − B ∗ ∈ Γ(Hom C ( ¯ F , ¯ E )) has the same zeroes but with opposite signs. Proof.
The first two statements follow immediately from the fact that − B ∗ is the composition of the canonical conjugation map ¯ F → F with the complex-linear bundle map − β † : F → ¯ E . For the third, it suffices to compare what β ∈ Γ(Hom C ( ¯ E, F )) and − β † : Γ(Hom C ( F, ¯ E )) look like in local trivializations near azero: one is minus the complex conjugate of the other, hence their zeroes count withopposite signs. (cid:3) Chris Wendl
As a warmup for the general case, we now fill in the details of Taubes’s proof ofTheorem 5.4 in the case ˙Σ = T := C \ ( Z ⊕ i Z )and E = T × C , i.e. a trivial line bundle. In this case I ( D ) = χ ( T ) + 2 c ( E ) = 0,so our aim is to prove ind( D ) = 0. What we will show in fact is that D is homotopicthrough a continuous family of Fredholm operators to one that is an isomorphism.Since E and F are now both trivial, it will suffice to consider the operator D := ¯ ∂ = ∂ s + i∂ t : H ( T , C ) → L ( T , C ) , whose formal adjoint is D ∗ := − ∂ = − ∂ s + i∂ t . An antilinear zeroth-order pertur-bation is then equivalent to a choice of function β : T → C , giving rise to a familyof operators D σ η := ¯ ∂η + σβ ¯ η for σ ∈ R , where ¯ η : T → C now denotes the straightforward complex conjugateof η . Let us assume that β : T → C is nowhere zero; note that this would not bepossible in more general situations, but is possible here because Hom C ( ¯ E, F ) is atrivial bundle.
Lemma . D σ is injective for all σ > sufficiently large. Proof.
Elliptic regularity implies any η ∈ ker D σ is smooth, so we shall restrictour attention to smooth functions η : T → C . We start by comparing the twosecond-order differential operators D ∗ D and D ∗ σ D σ : C ∞ ( T , C ) → C ∞ ( T , C ) . Both are nonnegative L -symmetric operators, and in fact the first is simply theLaplacian D ∗ D = − ∂ ¯ ∂ = ( − ∂ s + i∂ t )( ∂ s + i∂ t ) = − ∂ s − ∂ t = − ∆ . The formal adjoint of D σ takes the form D ∗ σ η = D ∗ η + σB ∗ η = D ∗ η + σβ ¯ η, thus for any η ∈ C ∞ ( T , C ), D ∗ σ D σ η = ( D ∗ + σB ∗ )( D + σB ) η = D ∗ D η + σ (cid:16) β ¯ ∂η − ∂ ( β ¯ η ) (cid:17) + σ B ∗ Bη = D ∗ D η + σ ( β∂ ¯ η − ( ∂β )¯ η − β∂ ¯ η ) + σ B ∗ Bη = D ∗ D η + σ B ∗ Bη − σ ( ∂β )¯ η. (5.2)This is a Weitzenb¨ock formula : its main message is that the Laplacian D ∗ D andthe related operator D ∗ σ D σ differ from each other only by a zeroth-order term that ectures on Symplectic Field Theory will be positive definite if σ is sufficiently large. Indeed, since β is nowhere zero, wehave | Bη | ≥ c | η | for some constant c >
0, thus k D σ η k L = h η, D ∗ σ D σ η i L = h η, D ∗ D η i L + σ h η, B ∗ Bη i L − σ h η, ( ∂β )¯ η i L = k D η k L + σ k Bη k L − σ h η, ( ∂β )¯ η i L ≥ (cid:0) σ c − σ k ∂β k C (cid:1) k η k L . We conclude that as soon as σ > D σ η cannot vanish unless k η k L = 0. (cid:3) Proof of Theorem 5.4 for E = T × C . The lemma above shows that onecan add a large antilinear perturbation to D = ¯ ∂ making the deformed operator D σ injective. By Lemma 5.11, the same argument applies to the formal adjoint D ∗ , implying that for sufficiently large σ > D ∗ σ is injective and thus D σ is alsosurjective, and therefore an isomorphism. This proves ind( D ) = ind( D σ ) = 0. (cid:3) Let’s consider which particular details of the setup made the proof above possible.First, the zeroth-order perturbation is complex antilinear. We used this, if onlyimplicitly, in deriving the Weitzenb¨ock formula (5.2): the key step is in the thirdline, where the two terms involving ∂ ¯ η cancel each other out and leave nothing butzeroth-order terms remaining. This would not have happened if e.g. B : E → F had been complex linear—we would then have seen terms depending on the firstderivative of η in D ∗ σ D σ η − D ∗ D η , and this would have killed the whole argument.The fact that this cancellation happens when the perturbation is antilinear probablylooks like magic at this point, but there is a principle behind it; we will discuss itfurther in § β : T → C is nowhere zero, inorder to obtain the lower bound on k Bη k L in terms of k η k L . This cannot alwaysbe achieved—it is possible in this special case only because E and F are bothtrivial bundles and thus so is Hom C ( ¯ E, F ). On more general bundles, the best wecould hope for would be to pick β ∈ Γ(Hom C ( ¯ E, F )) with finitely many zeroes, allnondegenerate. In this case the above argument fails, but it still tells us something.Suppose Σ ǫ ⊂ T is a region disjoint from the isolated zeroes of β . Then there existsa constant c ǫ >
0, dependent on the region Σ ǫ , such that k β ¯ η k L ( T ) ≥ k β ¯ η k L (Σ ǫ ) ≥ c ǫ k η k L (Σ ǫ ) , so instead of the estimate at the end of the proof above implying D σ is injective, weobtain one of the form k D σ η k L ( T ) ≥ c ǫ σ k η k L (Σ ǫ ) − cσ k η k L ( T ) . To see what this means, imagine we have sequences σ ν → ∞ and η ν ∈ ker D σ ν ,normalized so that k η ν k L = 1 for all ν . The estimate above then implies k η ν k L (Σ ǫ ) ≤ cc ǫ σ ν → ν → ∞ , so while all sections η ν have the same amount of “energy” (as measured via their L -norms), the energy is escaping from Σ ǫ as σ ν increases. This is true for any domainΣ ǫ disjoint from the zeroes, so we conclude that in the limit as σ → ∞ , sections in Chris Wendl ker D σ have their energy concentrated in infinitesimally small neighborhoods of thezeroes of β . We will see in the following how to extract useful information from thisconcentration of energy. The Weitzenb¨ock formula (5.2) can be generalized to a useful relation betweenany two Cauchy-Riemann type operators that differ by an antilinear zeroth-orderterm. To see this, we start with a short digression on holomorphic and antiholomor-phic vector bundles.A smooth function f : C ⊃ U → C is called antiholomorphic if it satisfies( ∂ s − i∂ t ) f = 0, which means its differential anticommutes with the complex structureon C . The class of antiholomorphic functions is not closed under composition, butit is closed under products, hence one can define an antiholomorphic structure on a complex vector bundle to be a system of local trivializations for which alltransition maps are antiholomorphic. Given the standard correspondence betweenholomorphic structures and Cauchy-Riemann type operators, it is easy to establisha similar correspondence between aniholomorphic structures and (complex-linear) anti-Cauchy-Riemann type operators, i.e. those which satisfy D ( f η ) = ( ∂f ) η + f D η for all f ∈ C ∞ ( ˙Σ , C ), where ∂f := df − i df ◦ j ∈ Ω , ( ˙Σ). We’ve seen one importantexample of such an operator already: if D : Γ( E ) → Γ( F ) is complex linear, then − D ∗ is a complex-linear anti-Cauchy-Riemann operator on F and thus endows F with an antiholomorphic structure. Another natural example occurs naturally onconjugate bundles: if E has a holomorphic structure, then ¯ E inherits from this anantiholomorphic structure. This is immediate from the fact that f : C ⊃ U → C isholomorphic if and only if ¯ f : U → C is antiholomorphic. If D : Γ( E ) → Γ( F ) =Ω , ( ˙Σ , E ) is the corresponding complex-linear Cauchy-Riemann type operator on E , we shall denote the resulting anti-Cauchy-Riemann operator by D : Γ( ¯ E ) → Γ( ¯ F ) = Ω , ( ˙Σ , ¯ E ) , where by definition D ¯ η = D η . Exercise . Show that if X and Y are antiholomorphic vector bundles overthe same base, then X ⊗ Y and Hom C ( X, Y ) both naturally inherit antiholomorphicbundle structures such that the obvious Leibniz rules are satisfied.
Remark: theproof of this is exactly the same as for holomorphic bundles, one only needs tochange some signs.
Exercise . Suppose X and Y are complex vector bundles over the samebase, carrying real-linear anti-Cauchy-Riemann operators ∂ X and ∂ Y respectively.Show that H := Hom R ( X, Y ) then admits a real-linear anti-Cauchy-Riemann oper-ator ∂ H such that for all Φ ∈ Γ( H ) and η ∈ Γ( X ), ∂ Y (Φ η ) = ( ∂ H Φ) η + Φ( ∂ X η ) . ectures on Symplectic Field Theory Hint: write ∂ X and ∂ Y as complex-linear operators with real-linear zeroth-orderperturbations, and apply Exercise 5.13. Show moreover that any C k -bounds satisfiedby the zeroth-order terms in ∂ X and ∂ Y are inherited by the zeroth-order term in ∂ H . The setup for the next result is as follows. We assume again m = 1, so E and F are line bundles. Fix β ∈ Γ(Hom C ( ¯ E, F )), define B ∈ Γ(Hom C ( E, F )) by Bη := β ¯ η ,and use this to define the perturbed Cauchy-Riemann type operator D B := D + B : Γ( E ) → Γ( F ) , whose formal adjoint is D ∗ B = D ∗ + B ∗ with B ∗ λ := β † ¯ λ . Proposition . The second-order differential operators D ∗ D and D ∗ B D B on E are related by D ∗ B D B η = D ∗ D ∗ η + B ∗ Bη − ( ∂ H β )¯ η, where ∂ H is a real-linear anti-Cauchy-Riemann type operator on Hom C ( ¯ E, F ) . More-over, if β is C -bounded on ˙Σ , then ∂ H β is C -bounded. Proof.
We have real-linear anti-Cauchy-Riemann operators D and − D ∗ on ¯ E and F respectively, so Exercise 5.14 produces an operator ∂ H on Hom C ( ¯ E, F ) forwhich the Leibniz rule is satisfied. We can then write D ∗ B D B η = ( D ∗ + B ∗ )( D + B ) η = D ∗ D η + β † D η − ( − D ∗ )( β ¯ η ) + B ∗ Bη = D ∗ D η + β † D ¯ η − ( ∂ H β )¯ η − β D ¯ η + B ∗ Bη = D ∗ D η + B ∗ Bη − ( ∂ H β )¯ η + (cid:16) β † − β (cid:17) D ¯ η. Here β and β † are both viewed as complex-linear bundle maps ¯ F → E , the latter inthe obvious way, and the former acting as ⊗ β on ¯ F = Λ , T ∗ ˙Σ ⊗ ¯ E with targetΛ , T ∗ ˙Σ ⊗ F = Λ , T ∗ ˙Σ ⊗ Λ , T ∗ ˙Σ ⊗ E = E . Choosing unitary local trivializations, β and β † are represented by the same complex-valued function: indeed, the latteris the transpose of the former as m -by- m complex matrices, but since m = 1, thismeans they are identical.Finally, we observe that the asymptotic convergence conditions satisfied by D on the cylindrical ends imply similar conditions for all other Cauchy-Riemann andanti-Cauchy-Riemann operators in this picture, yielding an estimate of the form k ∂ H β k C ≤ c k β k C globally on ˙Σ. (cid:3) Remark . The above proof used the assumption m = 1 in order to conclude β † − β ≡
0. For higher rank bundles, this imposes a nontrivial condition that mustbe satisfied in order for the Weitzenb¨ock formula to hold, cf. [ GW ]. Remark . We can now pick out a geometric reason for the miraculous can-cellation in the Weitzenb¨ock formula: the perturbation B is described by a complexbundle map ¯ E → F , where ¯ E and F both have natural antiholomorphic bun-dle structures defined via the complex-linear parts of D and − D ∗ respectively. Acomplex-linear perturbation B : E → F would not work because E is holomorphicrather than antiholomorphic: while D can be fit into the same Leibniz rule with − D ∗ , the same is not true of D . Chris Wendl
We continue in the setting of Proposition 5.15 and set D σ := D + σB : Γ( E ) → Γ( F )for σ >
0. After a compact perturbation of D , we can without loss of generalityalso impose the following assumptions on D , β ∈ Γ(Hom C ( ¯ E, F )) and the areaform d vol:(i) All zeroes of β are nondegenerate.(ii) Both | β | and 1 / | β | are bounded outside of a compact subset of ˙Σ.(iii) Near each point ζ ∈ ˙Σ with β ( ζ ) = 0, there exists a neighborhood D ( ζ ) ⊂ ˙Σof ζ , a holomorphic coordinate chart identifying ( D ( ζ ) , j, ζ ) with the unitdisk ( D , i, E over D ( ζ ) that identifies D with¯ ∂ = ∂ s + i∂ t : C ∞ ( D , C ) → C ∞ ( D , C ) and β with one of the functions β ( z ) = z or β ( z ) = ¯ z, the former if ζ is a positive zero and the latter if it is negative.(iv) In the holomorphic coordinate on D ( ζ ) described above, d vol is the stan-dard Lebesgue measure.As in the torus case discussed in § β for sections η ∈ ker D σ as σ → ∞ . To understand what really happens in this limit, we will use a rescalingtrick. Denote the zero set of β by Z ( β ) = Z + ( β ) ∪ Z − ( β ) ⊂ ˙Σ , partitioned into the positive and negative zeroes. For any η ∈ Γ( E ), ζ ∈ Z ± ( β ) and σ >
0, we then define a rescaled function η ( ζ,σ ) : D √ σ → C : z √ σ η ( z/ √ σ ) , where the right hand side denotes the local representation of η on D ( ζ ) in the chosencoordinate and trivialization. Notice that the equation D σ η = 0 appears in this localrepresentation as either ¯ ∂η + σz ¯ η = 0 or ¯ ∂η + σ ¯ z ¯ η = 0 depending on the sign of ζ ,and the function f := η ( ζ,σ ) then satisfies¯ ∂f + z ¯ f = 0 or ¯ ∂f + ¯ z ¯ f = 0 on D √ σ . We will take a closer look at these two PDEs in § (cid:13)(cid:13) η ( ζ,σ ) (cid:13)(cid:13) L ( D √ σ ) = k η k L ( D ( ζ )) . Lemma . Assume σ ν → ∞ , and η ν ∈ ker D σ ν is a sequence satisfying auniform L -bound. Then after passing to a subsequence, the rescaled functions η ζν := η ( ζ,σ ν ) ν : D √ σ ν → C for each ζ ∈ Z ± ( β ) converge in C ∞ loc ( C ) to smooth functions ectures on Symplectic Field Theory η ζ ∞ ∈ L ( C ) satisfying ¯ ∂η ζ ∞ + zη ζ ∞ = 0 if ζ ∈ Z + ( β ) , ¯ ∂η ζ ∞ + ¯ zη ζ ∞ = 0 if ζ ∈ Z − ( β ) . Moreover, if ξ ν ∈ ker D σ ν is another sequence with these same properties and con-vergence ξ ζν → ξ ζ ∞ , then lim ν →∞ h η ν , ξ ν i L ( E ) = X ζ ∈ Z ( β ) h η ζ ∞ , ξ ζ ∞ i L ( C ) . Proof.
The uniform L -bound implies uniform bounds on k η ζν k L ( D R ) for every R >
0, where ν here is assumed sufficiently large so that R < √ σ ν . Since η ζν satisfiesa Cauchy-Riemann type equation on D R , the usual elliptic estimates (see Lecture 2)then imply uniform H k -bounds for every k ∈ N on every compact subset in theinterior of D R , hence η ζν has a C ∞ loc -convergent subsequence on C , and the limit η ζ ∞ clearly satisfies the stated PDE. The uniform L -bound also implies a uniform boundon k η ζν k L ( D √ σν ) and thus an R -independent uniform bound on k η ζν k L ( D R ) as ν → ∞ ,implying that η ζ ∞ is in L ( C ).The limit of h η ν , ξ ν i L ( E ) is now proved using the Weitzenb¨ock formula. Let˙Σ ǫ := ˙Σ \ [ ζ ∈ Z ( β ) D ( ζ ) , so there exists a constant c > β satisfies | β ( z )¯ v | ≥ c | v | for all v ∈ E z , z ∈ ˙Σ ǫ . (Note that this depends on the assumption of 1 / | β | being bounded outsideof a compact subset.) Now by Proposition 5.15,0 = k D σ ν η ν k L ( ˙Σ) = h η ν , D ∗ σ ν D σ ν η ν i L ( ˙Σ) = h η ν , D ∗ D η ν i L ( ˙Σ) + σ ν h η ν , B ∗ Bη ν i L ( ˙Σ) − σ ν h η ν , ( ∂ H β )¯ η ν i L ( ˙Σ) ≥ k D η ν k L ( ˙Σ) + σ ν c k η ν k L ( ˙Σ ǫ ) − σ ν c ′ k η ν k L ( ˙Σ) ≥ σ ν c k η ν k L ( ˙Σ ǫ ) − σ ν c ′ k η ν k L ( ˙Σ) for some constant c ′ > ν . This implies k η ν k L ( ˙Σ ǫ ) ≤ c ′ c σ ν k η ν k L ( ˙Σ) → ν → ∞ since k η ν k L ( ˙Σ) is uniformly bounded. The same estimate applies to ξ ν , so that h η ν , ξ ν i L ( ˙Σ ǫ ) → ν →∞ h η ν , ξ ν i L ( ˙Σ) = lim ν →∞ X ζ ∈ Z ( β ) h η ν , ξ ν i L ( D ( ζ )) = lim ν →∞ X ζ ∈ Z ( β ) h η ζν , ξ ζν i L ( D √ σν ) = X ζ ∈ Z ( β ) h η ζ ∞ , ξ ζ ∞ i L ( C ) . (cid:3) Chris Wendl
The rescaling trick in the previous section produced smooth solutions f : C → C of class L ( C ) to the two equations¯ ∂f + z ¯ f = 0 , ¯ ∂f + ¯ z ¯ f = 0 . It turns out that we can say precisely what all such solutions are. Write D + f := ¯ ∂f + z ¯ f and D − f := ¯ ∂f + ¯ z ¯ f . Both operators differ from ¯ ∂ by antilinear perturbations,so they satisfy Weitzenb¨ock formulas relating D ∗± D ± to the Laplacian − ∆ = ¯ ∂ ∗ ¯ ∂ = − ∂ s − ∂ t . Indeed, repeating Proposition 5.15 in these special cases gives D ∗ + D + f = − ∆ f + | z | f − f and D ∗− D − f = − ∆ f + | z | f. To make use of this, recall that a smooth function u : U → R on an open subset U ⊂ C is called subharmonic if it satisfies − ∆ u ≤ . Subharmonic functions satisfy a mean value property : − ∆ u ≤ U ⇒ u ( z ) ≤ πr Z D r ( z ) u ( z ) dµ ( z ) for all D r ( z ) ⊂ U , where D r ( z ) ⊂ C denotes the disk of radius r > z ∈ U , and dµ ( z )is the Lebesgue measure on C ; see e.g. [ Eva98 , p. 85].
Exercise . Show that for any smooth complex-valued function f on anopen subset of C , ∆ | f | = 2 Re h f, ∆ f i + 2 |∇ f | , where h , i denotes the standard Hermitian inner product on C and |∇ f | := | ∂ s f | + | ∂ t f | . Proposition . The equation ¯ ∂f + ¯ z ¯ f = 0 does not admit any nontrivialsmooth solutions f ∈ L ( C , C ) . Proof. If f : C → C is smooth with D − f = 0, then the Weitzenb¨ock formulafor D − implies ∆ f = | z | f . Then by Exercise 5.19,∆ | f | = 2 Re h f, | z | f i + 2 |∇ f | = 2 | z | | f | + 2 |∇ f | , implying that | f | : C → R is subharmonic. Now if f ( z ) = 0 for some z ∈ C , themean value property implies Z D r ( z ) | f ( z ) | dµ ( z ) ≥ πr | f ( z ) | → ∞ as r → ∞ , so f L ( C ). (cid:3) Proposition . Every smooth solution f ∈ L ( C , C ) to the equation ¯ ∂f + z ¯ f = 0 is a constant real multiple of f ( z ) := e − | z | . ectures on Symplectic Field Theory Proof.
We claim first that every smooth solution in L ( C , C ) of D + f = 0 ispurely real valued. The Weitzenb¨ock formula for this case gives ∆ f = | z | f − f , andtaking the difference between this equation and its complex conjugate then impliesthat u := Im f : C → R satisfies ∆ u = ( | z | + 2) u. Now by Exercise 5.19, ∆( u ) = 2 |∇ u | + 2( | z | + 2) u ≥ , so u : C → R is subharmonic, and the mean value property implies as in the proofof Prop. 5.20 that u L ( C ) and hence f L ( C ) unless u ≡
0. This proves theclaim.It is easy to check however that f is a solution and is in L ( C ). Since it is alsonowhere zero, every other solution f must then take the form f ( z ) = v ( z ) f ( z ) forsome real-valued function v : C → R . Since D + is a Cauchy-Riemann type operator,the Leibniz rule then implies ¯ ∂v ≡
0. But the only globally holomorphic functionswith trivial imaginary parts are constant. (cid:3)
Now we’re getting somewhere.
Lemma . Suppose the assumptions of § β ∈ Γ(Hom C ( ¯ E, F )) has I + ≥ positive and I − ≥ negative zeroes. Then for all σ > sufficiently large, dim ker D σ ≤ I + and dim coker D σ ≤ I − . In particular, for sufficiently large σ , D σ is injective if all zeroes of β are negativeand surjective if all zeroes are positive. Proof.
Arguing by contradiction, suppose there exists a sequence σ ν → ∞ suchthat dim ker D σ ν > I + , and pick ( I + +1) sequences of sections η ν , . . . , η I + +1 ν ∈ ker D σ ν which form L -orthonormal sets for each ν . By Lemma 5.18, we can then extracta subsequence such that rescaling near the zeroes of β produces C ∞ loc -convergentsequences whose limits form an ( I + + 1)-dimensional orthonormal set in M ζ ∈ Z ( β ) L ( C , C ) , where the component functions f ∈ L ( C , C ) for ζ ∈ Z + ( ζ ) satisfy ¯ ∂f + z ¯ f = 0,while those for ζ ∈ Z − ( ζ ) satisfy ¯ ∂f + ¯ z ¯ f = 0. Proposition 5.20 now implies thatthe component functions for ζ ∈ Z − ( ζ ) are all trivial, and by Proposition 5.21,the components for ζ ∈ Z + ( ζ ) belong to 1-dimensional subspaces ker D + ⊂ L ( C )generated by the function e − | z | . We conclude that the limiting orthonormal setlives in a precisely I + -dimensional subspace M ζ ∈ Z + ( β ) ker D + ⊂ M ζ ∈ Z ( β ) L ( C , C ) , and this is a contradiction since there are I + + 1 elements in the set. Chris Wendl
Applying the same argument to the formal adjoint implies similarly dim ker D ∗ σ ≤ I − for σ sufficiently large. (cid:3) We would next like to turn the two inequalities in the above lemma into equal-ities, which means showing that the I + -dimensional subspace of L ζ ∈ Z + ( β ) L ( C , C )generated by solutions of ¯ ∂f + z ¯ f = 0 is isomorphic to ker D σ for σ sufficiently large.This requires a simple example of a linear gluing argument, the point of which isto reverse the “convergence after rescaling” process that we saw in Lemma 5.18.The first step is a pregluing construction which turns elements of L ζ ∈ Z + ( β ) ker D + into approximate solutions to D σ η = 0 for large σ . To this end, fix a smooth bumpfunction ρ ∈ C ∞ (˚ D , [0 , , ρ | D / ≡ ζ ∈ Z + ( β ) and σ > ζσ : ker D + → Γ( E )such that Φ ζσ ( f ) is a section with support in D ( ζ ) whose expression in our fixedcoordinate and trivialization on that neighborhood is the function f ζσ ( z ) = ρ ( z ) √ σf ( √ σz ) . Adding up the Φ ζσ for all ζ ∈ Z + ( β ) then produces a linear mapΦ σ : M ζ ∈ Z + ( β ) ker D + → Γ( E )whose image consists of sections supported near Z + ( β ), each a linear combinationof cut-off Gaussians with energy concentrated in smaller neighborhoods of Z + ( β )for larger σ . These sections are manifestly not in ker D σ since they vanish on opensubsets and thus violate unique continuation, but they are close, in a quantitativesense: Lemma . For each σ > , there exists a constant c σ > such that k D σ Φ σ ( f ) k L ≤ c σ k f k L for all f ∈ M ζ ∈ Z + ( β ) ker D + , and c σ → as σ → ∞ . Moreover, for every pair f, g ∈ L ζ ∈ Z + ( β ) ker D + , h Φ σ ( f ) , Φ σ ( g ) i L → h f, g i L as σ → ∞ . Proof.
First, observe that any f ∈ L ζ ∈ Z + ( β ) ker D + is described by a collectionof functions { f ζ ∈ L ( C ) } ζ ∈ β + ( Z ) which take the form f ζ ( z ) = K ζ e − | z | , ectures on Symplectic Field Theory for some constants K ζ ∈ R . Since each f ζ is in ker D + , we find D σ (cid:0) Φ σ ( f ) | D ( ζ ) (cid:1) ( z ) = ∂ρ ( z ) √ σf ζ ( √ σz ) + ρ ( z ) σ∂f ζ ( √ σz )+ σzρ ( z ) √ σf ζ ( √ σz )= ∂ρ ( z ) √ σf ζ ( √ σz ) + ρ ( z ) σ ( D + f ζ )( √ σz )= ∂ρ ( z ) √ σK ζ e − σ | z | . (5.3)Now since ∂ρ = 0 in D / , we obtain k D σ Φ σ ( f ) k L = X ζ ∈ Z + ( β ) Z D ( ζ ) | D σ Φ σ ( f )( z ) | dµ ( z )= X ζ ∈ Z + ( β ) Z D \ D / | ∂ρ ( z ) | σK ζ e − σ | z | dµ ( z ) ≤ Iσe − σ/ X ζ ∈ Z + ( β ) K ζ , where we abbreviate I := R D \ D / (cid:12)(cid:12) ¯ ∂ρ ( z ) (cid:12)(cid:12) dµ ( z ). The norm of f is given by k f k L = X ζ ∈ Z + ( β ) Z C K ζ e −| z | dµ ( z ) = (cid:18)Z C e −| z | dµ ( z ) (cid:19) X ζ ∈ Z + ( β ) K ζ . We conclude that there is a bound of the form k D σ Φ σ ( f ) k L ≤ C √ σe − σ/ k f k L , which proves the first statement since √ σe − σ/ → σ → ∞ .The second statement follows by a change of variable, since h Φ σ ( f ) , Φ σ ( g ) i L = X ζ ∈ Z + ( β ) h Φ σ ( f ) | D ( ζ ) , Φ σ ( g ) | D ( ζ ) i L ( D ( ζ )) = X ζ ∈ Z + ( β ) Z D ρ ( z ) σf ζ ( √ σz ) g ζ ( √ σz ) dµ ( z )= X ζ ∈ Z + ( β ) Z D √ σ ρ (cid:18) z √ σ (cid:19) f ζ ( z ) g ζ ( z ) dµ ( z )The functions f ζ and g ζ are both real multiples of e − | z | , so this last integral for each ζ ∈ Z + ( β ) is bounded between R D √ σ/ f ζ ( z ) g ζ ( z ) dµ ( z ) and R D √ σ f ζ ( z ) g ζ ( z ) dµ ( z ),both of which converge to R C f ζ ( z ) g ζ ( z ) dµ ( z ) as σ → ∞ , thuslim σ →∞ h Φ σ ( f ) , Φ σ ( g ) i L = h f, g i L . (cid:3) To turn approximate solutions into actual solutions, letΠ σ : L ( E ) → ker D σ denote the orthogonal projection. We will prove: Chris Wendl
Proposition . If all zeroes of β are positive, then the linear map Π σ ◦ Φ σ : M ζ ∈ Z + ( β ) ker D + → ker D σ is injective for all σ > sufficiently large. This statement says in effect that whenever σ > η :=Φ σ ( f ) ∈ Γ( E ) is in the image of the pregluing map, with f normalized by k f k L = 1,we can find a “correction” ξ ∈ (ker D σ ) ⊥ such that η + ξ = 0 but D σ ( η + ξ ) = 0 . An element ξ ∈ (ker D σ ) ⊥ with the second property certainly exists, and in factit’s unique: indeed, the assumption Z − ( β ) = ∅ implies via Lemma 5.22 that D σ issurjective and thus restricts to an isomorphism from (ker D ) ⊥ ∩ H ( E ) to L ( F ),with a bounded right inverse Q σ : L ( F ) → H ( E ) ∩ (ker D ) ⊥ , hence ξ := − Q σ ( D σ η ). We know moreover from Lemma 5.23 that k η k L is closeto k f k L = 1, so to prove η + ξ = 0, it would suffice to show k ξ k L is small, whichsounds likely since we also know k D σ η k L is small and Q σ is a bounded operator. Tomake this reasoning precise, we just need to have some control over k Q σ k as σ → ∞ ,or equivalently, a quantitative measure of the injectivity of D σ | (ker D σ ) ⊥ ∩ H ( E ) . Thisrequires one last appeal to the Weitzenb¨ock formula. Lemma . Assume all zeroes of β are positive. Then there exist constants c > and σ such that for all σ > σ , k η k L ≤ c k D σ η k L for all η ∈ H ( E ) ∩ (ker D σ ) ⊥ . Proof.
Let us instead prove that if zeroes of β are all negative , then the samebound holds for all η ∈ H ( E ). The stated result follows from this by consideringthe formal adjoint and using Exercise 5.26 below. Note that by density, it sufficesto prove the estimate holds for all η ∈ C ∞ ( E ).Assume therefore that Z + ( β ) = ∅ and, arguing by contradiction, suppose thereexist sequences σ ν → ∞ and η ν ∈ C ∞ ( E ) with k η ν k L = 1 and k D σ ν η ν k L → . The usual rescaling trick and application of the Weitzenb¨ock formula then producesfor each ζ ∈ Z − ( β ) a sequence of functions η ζν := η ( ζ,σ ν ) ν : D √ σ ν → C which satisfy X ζ ∈ Z − ( β ) k η ζν k L ( D √ σν ) → k D − η ζν k L ( D √ σν ) → ν → ∞ . Indeed, defining ˙Σ ǫ as in the proof of Lemma 5.18, a similar applicationof the Weitzenb¨ock formula yields k D σ ν η ν k L ( ˙Σ) ≥ σ ν c k η ν k L ( ˙Σ ǫ ) − σ ν c ′ k η ν k L ( ˙Σ) = σ ν c k η ν k L ( ˙Σ ǫ ) − σ ν c ′ , ectures on Symplectic Field Theory for some c ′ >
0. Thus we obtain k η ν k L ( ˙Σ ǫ ) ≤ k D σ ν η ν k L ( ˙Σ) c σ ν + c ′ σ ν c → ν → ∞ , so there is again concentration of energy near the zeroes of the antilinear perturba-tion: in particular, 1 = lim ν →∞ k η ν k L ( ˙Σ) = lim ν →∞ k η ν k L ( ˙Σ ǫ ) + lim ν →∞ X ζ ∈ Z − ( β ) k η ν k L ( D ( ζ )) = lim ν →∞ X ζ ∈ Z − ( β ) k η ζν k L ( D √ σν ) . Moreover, we have D − η ζν ( z ) = 1 σ ν ∂η ν (cid:18) z √ σ ν (cid:19) + ¯ z √ σ ν ¯ η ν (cid:18) z √ σ ν (cid:19) = 1 σ ν D σ ν η ν (cid:18) z √ σ ν (cid:19) . Taking the square of the norms on each side, we may integrate and use change ofvariables to obtain k D − η ζν k L ( D √ σν ) = 1 √ σ ν k D σ ν η ν k L ( D ( ζ )) → ν → ∞ . The elliptic estimates from Lecture 2 now provide uniform H k -bounds for each η ζν on compact subsets of C for every k ∈ N , so that a subsequence converges in C ∞ loc ( C )to a smooth map η ζ ∞ ∈ L ( C , C ) satisfying D − η ζ ∞ = 0. But P ζ ∈ Z − ( β ) k η ζ ∞ k L ( C ) = 1,so at least one of these solutions is nontrivial and thus contradicts Proposition 5.20. (cid:3) Exercise . Show that for any Fredholm Cauchy-Riemann type operator D on E , the following two estimates are equivalent, with the same constant c > k η k L ( E ) ≤ c k D η k L ( F ) for all η ∈ H ( E ) ∩ (ker D ) ⊥ ;(ii) k λ k L ( F ) ≤ c k D ∗ λ k L ( E ) for all λ ∈ H ( F ) ∩ (ker D ∗ ) ⊥ . Hint: elliptic regularity implies that for D and D ∗ as bounded linear operators H → L , (ker D ) ⊥ = im D ∗ and (ker D ∗ ) ⊥ = im D . Proof of Proposition 5.24.
If the statement is not true, then there existsequences σ ν → ∞ and f ν ∈ M ζ ∈ Z + ( β ) ker D + such that k f ν k L = 1 and η ν := Φ σ ν ( f ν ) ∈ (ker D σ ν ) ⊥ for all ν . Lemmas 5.23and 5.25 then provide estimates of the form • k η ν k L → • k D σ ν η ν k L →
0, and • k η ν k L ≤ c k D σ ν η ν k L Chris Wendl as ν → ∞ , with c > ν . These imply:1 = lim ν →∞ k η ν k L ≤ lim ν →∞ c k D σ ν η ν k L = 0 . (cid:3) We’ve proved:
Proposition . Suppose the assumptions of § β ∈ Γ(Hom C ( ¯ E, F )) has I + ≥ positive and I − ≥ negative zeroes. If I − = 0 , then D σ is surjective with dim ker D σ = I + for all σ > sufficiently large. If I + = 0 ,then D σ is injective with dim coker D σ = I − for all σ > sufficiently large. In eithercase, ind( D σ ) = I + − I − for all σ > sufficiently large. (cid:3) Proposition 5.27 suffices to prove the index formula in the closed case, but thereis an additional snag if Γ = ∅ : since H ( ˙Σ) ֒ → L ( ˙Σ) is not a compact inclusion,we have no guarantee that D and D σ := D + σB will have the same index, andgenerally they will not. A solution to this problem has been pointed out by ChrisGerig [ Ger ], using a special class of asymptotic operators that also originate in thework of Taubes (see [
Tau10 , Lemma 2.3]).In general, the only obvious way to guarantee ind( D ) = ind( D σ ) for large σ > every operator in the family { D σ } σ ≥ to be Fredholm, whichis not automatic since the zeroth-order perturbation B : E → F is required to bebounded away from zero near ∞ and must therefore change the asymptotic operatorsat the punctures. We are therefore led to ask: Question.
For what nondegenerate asymptotic operators A : H ( E ) → L ( E ) on a Hermitian line bundle ( E, J, ω ) → S can one find complex-antilinear bundlemaps B : E → E such that A σ := A − σB : H ( E ) → L ( E ) is an isomorphism for every σ ≥ ? It turns out that it will suffice to find, for each unitary trivialization σ and every k ∈ Z , a particular pair ( A k , B k ) such that A k − σB k is nondegenerate for all σ ≥ µ τ CZ ( A k ) = k . To see why, let us proceed under the assumption that such pairscan be found, and use them to compute the index: Lemma . Given D as in Theorem 5.4, fix asymptotic trivializations τ andsuppose that for each puncture z ∈ Γ there exists an asymptotic operator A ′ z on ( E z , J z , ω z ) with µ τ CZ ( A ′ z ) = µ τ CZ ( A z ) , such that if A ′ z is written with respect to τ as − J ∂ t − S z ( t ) , then the deformed asymptotic operator (5.4) C ∞ ( S , R ) → C ∞ ( S , R ) : η
7→ − J ∂ t η − S z ( t ) η − σβ z ( t )¯ η ectures on Symplectic Field Theory is nondegenerate for some loop β z : S → C \ { } and every σ ≥ . Then ind( D ) = χ ( ˙Σ) + 2 c τ ( E ) + X z ∈ Γ + wind( β z ) − X z ∈ Γ − wind( β z ) . Proof.
Since µ τ CZ ( A z ) = µ τ CZ ( A ′ z ), we can deform A z to A ′ z continuouslythrough a family of nondegenerate asymptotic operators. It follows that we candeform D through a continuous family of Fredholm Cauchy-Riemann type oper-ators to a new operator D ′ whose asymptotic operators are A ′ z for z ∈ Γ, andind( D ′ ) = ind( D ). We are free to assume in fact that D ′ is written with respect tothe trivialization τ on the cylindrical end near z ∈ Γ ± as ∂ s + J ∂ t + S z ( t ) . Now choose β ∈ Γ(Hom C ( ¯ E, F )) with nondegenerate zeroes such that the deformedoperators D σ η := D ′ η + σβ ¯ η appear in trivialized form on the cylindrical end near z ∈ Γ ± as D σ η = ∂ s η + J ∂ t η + S z ( t ) η + σβ z ( t )¯ η. This means D σ is asymptotic at z to (5.4), which is nondegenerate for every σ ≥ D σ is Fredholm for every σ ≥ D ) = ind( D σ ) . The trivializations τ induce trivializations over the cylindrical ends for ¯ E and F = Λ , T ∗ ˙Σ ⊗ E , and the expression for β in the resulting asymptotic trivializationof Hom C ( ¯ E, F ) near z ∈ Γ is β z ( t ). It follows that the signed count of zeroes of β is i ( D ) := c τ (Hom C ( ¯ E, F )) + X z ∈ Γ + wind( β z ) − X z ∈ Γ − wind( β z )= χ ( ˙Σ) + 2 c τ ( E ) + X z ∈ Γ + wind( β z ) − X z ∈ Γ − wind( β z ) , where the computation c τ (Hom C ( ¯ E, F )) = χ ( ˙Σ) + 2 c τ ( E ) follows from the naturalisomorphismHom C ( ¯ E, F ) = ¯ E ∗ ⊗ F = E ⊗ F = E ⊗ Λ , T ∗ ˙Σ ⊗ E = Λ , T ∗ ˙Σ ⊗ E ⊗ E = T ˙Σ ⊗ E ⊗ E. We are free to assume that all zeroes of β are either positive or negative, dependingon the sign of i ( D ). Proposition 5.27 then implies ind( D σ ) = i ( D ) for large σ . (cid:3) Notice that instead of nondegenerate families A − σB parametrized by σ ∈ [0 , ∞ ),it is just as well to find such families which are nondegenerate and have the rightConley-Zehnder index for all σ >
0, as the σ ≥ A − B ) − σB for σ ≥
0. The following lemma thus completes the proofof Theorem 5.4.
Lemma . For every k ∈ Z , the trivial Hermitian line bundle over S admitsan asymptotic operator A k and a loop β k : S → C \ { } such that the deformedasymptotic operators A k,σ η := A k η − σβ k ¯ η Chris Wendl are nondegenerate for every σ > and satisfy µ CZ ( A k,σ ) = wind( β k ) = k. Proof.
We claim that the choices A k η := − J ∂ t η − πkη and β k ( t ) := e πikt do the trick. We prove this in three steps. Step 1: k = 0 . The above formula gives A ,σ = − J ∂ t η − σ ¯ η , in which the σ = 1 case is precisely the operator that we used in Lecture 3 to normalize theConley-Zehnder index, hence µ CZ ( A , ) = 0 by definition. More generally, all ofthese operators can be expressed in the form A := − J ∂ t − S where S ∈ End R ( R )is a constant nonsingular 2-by-2 symmetric matrix that anticommutes with J . Weclaim that all asymptotic operators of this form are nondegenerate. Indeed, theconditions S T = S and SJ = − J S for J = (cid:18) −
11 0 (cid:19) imply that S takes the form (cid:18) a bb − a (cid:19) with det S = − a − b = 0, and moreover S is of this form if and only if J S also is. In particular, J S is traceless, symmetric, and nonsingular. Solutionsof A η = 0 then satisfy ˙ η = J Sη , which has no periodic solutions since J S has onepositive and one negative eigenvalue, hence ker A = { } . Step 2: even k . There is a cheap trick to deduce the case k = 2 m for any m ∈ N from the k = 0 case. Recall that by Exercise 3.37 in Lecture 3, conjugating A ,σ bya change of trivialization changes its Conley-Zehnder index by twice the degree ofthat change. In particular, the operator˜ A ,σ η := e πimt A ,σ ( e − πimt η )is also a nondegenerate asymptotic operator, but with µ CZ ( ˜ A ,σ ) = µ CZ ( A ,σ )+2 m = k . Explicitly, we compute˜ A ,σ η = − J ∂ t η − πkη − σke πikt ¯ η, so A k,σ = ˜ A ,σ/k is also nondegenerate for every σ > Step 3: odd k . Another cheap trick relates each A k,σ to A k,σ after an adjustmentin σ . Given an arbitrary asymptotic operator A = − J ∂ t − S ( t ) and m ∈ N , define A m := − J ∂ t − mS ( mt ) . Geometrically, if A is a trivialized representation for the asymptotic operator ofa Reeb orbit γ : S → M , then A m is the operator for the m -fold covered orbit γ m : S → M : t γ ( mt ). It is easy to check in particular that if we define η m ( t ) := η ( mt ) for any given loop η : S → R , then A m η m = m ( A η ) m , so this gives an embedding of ker A into ker A m , implying that whenever A m isnondegenerate for some m ∈ N , so is A . To make use of this, observe that A k,σ η = − J ∂ t η − π kη − σe πikt ¯ η = A k, σ η, so A k,σ is nondegenerate for all σ > A k,σ . (cid:3) ectures on Symplectic Field Theory The proof of Theorem 5.4 is now complete.
Exercise . Derive a Weitzenb¨ock formula for asymptotic operators and useit to show that for any asymptotic operator A on the trivial Hermitian line bundleand any smooth β : S → C \ { } , the deformed operators A σ η := A η − σβ ¯ η areall nondegenerate for σ > µ CZ ( A σ ) =wind( β ) for large σ > Symplectic cobordisms and moduli spacesContents
In this lecture we introduce the moduli spaces of holomorphic curves that areused to define SFT.
In Lecture 1, we motivated the notion of a contact manifold by consideringhypersurfaces M in a symplectic manifold ( W, ω ) that satisfy a convexity (also knownas “contact type”) condition. The point of that condition was that it presents M as one member of a smooth 1-parameter family of hypersurfaces that all have thesame Hamiltonian dynamics; that 1-parameter family furnishes the basic model ofwhat we call the symplectization of M with its induced contact structure. A usefulgeneralization of this notion was introduced in [ HZ94 ] and was later recognized tobe the most natural geometric setting for punctured holomorphic curves. It has theadvantage of allowing us to view seemingly distinct theories such as HamiltonianFloer homology as special cases of SFT—and even if we are only interested in contactmanifolds, the generalization sometimes makes computations easier than they mightbe in a purely contact setting.Recall that every smooth hypersurface M in a 2 n -dimensional symplectic mani-fold ( W, ω ) has a characteristic line field ker ( ω | T M ) ⊂ T M, whose integral curves are the orbits on M of any Hamiltonian vector field generatedby a function H : W → R that has M as a regular level set. We say that M ⊂ ( W, ω )is stable if a neighborhood of M admits a stabilizing vector field V : this meansthat V is transverse to M and the 1-parameter family of hypersurfaces M t := ϕ tV ( M ) , − ǫ < t < ǫ generated by the flow ϕ tV of V has the property that each of the diffeomorphisms M → M t defined by flowing along V preserves characteristic line fields. Chris Wendl
Exercise . Show that if V is a stabilizing vector field for M ⊂ ( W, ω ), thenthe 2-form and 1-form pair (Ω , Λ) defined on M byΩ := ω | T M , Λ := ι V ω | T M has the following properties:(i) Ω | ker Λ is nondegenerate;(ii) ker Ω ⊂ ker d Λ.Show moreover that if M is assigned the orientation for which V is positively trans-verse to M and ξ := ker Λ ⊂ T M is assigned the natural co-orientation determinedby Λ, then the induced orientation of ξ matches the orientation determined by thesymplectic vector bundle structure Ω | ξ , hence condition (i) can equivalently be writ-ten as(iii) Λ ∧ Ω n − > W = 2 n .A stable Hamiltonian structure (or “SHS” for short) on an arbitrary oriented(2 n − M is a pair (Ω , Λ) consisting of a closed 2-form Ωand 1-form Λ such that properties (ii) and (iii) in Exercise 6.1 are satisfied.
Exercise . Show that if (Ω , Λ) is a stable Hamiltonian structure, then ω := d ( r Λ) + Ωis a symplectic form on ( − ǫ, ǫ ) × M for ǫ > r denotes thecoordinate on ( − ǫ, ǫ ); moreover, { } × M is a stable hypersurface in (( − ǫ, ǫ ) × M, ω ). Example . If M ⊂ ( W, ω ) is a contact type hypersurface, then a Liouvillevector field V transverse to M is a stabilizing vector field, and the induced stableHamiltonian structure is ( dα, α ), where α := λ | T M with λ := ω ( V, · ). We will referto this example henceforward as the contact case . Proposition . Suppose M ⊂ ( W, ω ) is a closed stable hypersurface withstabilizing vector field V and induced stable Hamiltonian structure (Ω , Λ) where Ω = ω | T M and
Λ = ι V ω | T M . Then a neighborhood of M in ( W, ω ) admits a sym-plectomorphism to (( − ǫ, ǫ ) × M, d ( r Λ) + Ω) for some ǫ > , identifying M ⊂ W with { } × M ⊂ ( − ǫ, ǫ ) × M . Proof.
By the smooth tubular neighbourhood theorem and the preceeding ex-ercise, we can view ω = d ( r Λ) + Ω as a symplectic form in some neighbourhood U ∼ = (( − ǫ, ǫ ) × M ) of M . In this neighbourhood,( ω − ω ) | M = 0by definition of ω and thus ω − ω = dµ for some 1-form µ such that µ | M = 0. Now define ω t = ω + t dµ ectures on Symplectic Field Theory and observe that it is a closed 2-form which can be assumed to be non-degeneratefor a small enough choice of U . Solving the Moser equation ι v t ω t = − µ yields a well-defined, time-dependent vector field v t with the property that v t | M = 0.Working back we produce an isotopy as follows: dι v t ω t = − dµ ⇒L v t ω t = dι v t ω t + ι v t dω t = dι v t ω t = − dµ = − dω t dt ⇒ ddt ( ρ ∗ t ω t ) = L v t ω t + dω t dt = 0where ρ ∗ t is the flow of v t . Then ρ ∗ t ω t = ρ ∗ ω = ω since ρ is the identity. The required symplectomorphism is then ρ : ρ − U → U and the fact that M is fixed under the isotopy follows from v t | M = 0. (cid:3) Example . In the contact case (Ω , Λ) = ( dα, α ), the symplectic form on thecollar neighborhood in Proposition 6.4 can be rewritten as d ( e t α ) by defining thecoordinate t := ln( r + 1). The proposition is easier to prove in this case: one canconstruct the collar neighborhood simply by flowing along V , with no need for theMoser isotopy trick.A stable Hamiltonian structure H = (Ω , Λ) gives rise to two important additionalobjects: a co-oriented hyperplane distribution ξ := ker Λ , and a positively transverse vector field R determined by the conditionsΩ( R, · ) ≡ R ) ≡ . By analogy with the contact case, we will refer to R as the Reeb vector field of H .The condition ker Ω ⊂ ker d Λ implies that it reduces to the usual contact notion ofthe Reeb vector field for Λ whenever the latter happens also to be a contact form.The symplectization of ( M, H ) for any stable Hamiltonian structure H =(Ω , Λ) can be defined by choosing suitable diffeomorphisms of ( − ǫ, ǫ ) × M with R × M : equivalently, this means we consider R × M with the family of symplecticforms ω ϕ defined by(6.1) ω ϕ := d ( ϕ ( r )Λ) + Ωwhere ϕ is chosen arbitrarily from the set(6.2) T := (cid:8) ϕ ∈ C ∞ ( R , ( − ǫ, ǫ )) (cid:12)(cid:12) ϕ ′ > (cid:9) . Chris Wendl
Example . The following stable Hamiltonian structure places HamiltonianFloer homology into the setting of SFT. Suppose (
W, ω ) is a closed symplectic man-ifold and H : S × W → R is a smooth function, and denote H t := H ( t, · ) : W → R .The time-dependent Hamiltonian vector field X t defined by dH t = − ω ( X t , · ) canthen be viewed as defining a symplectic connection on the trivial symplectic fiberbundle M := S × W t −→ S , i.e. the flow of R ( t, x ) := ∂ t + X t ( x ) defines symplectic parallel transport mapsbetween fibers. The horizontal subbundle for this connection is the “symplecticcomplement” of the vertical subbundle with respect to the closed 2-formΩ = ω + dt ∧ dH. In other words, Ω restricts to the fibers of M → S as ω and the subbundle { X ∈ T M | ω ( X, · ) | T ( { const }× W ) } is generated by R , so Ω is the connection -form defining the connection, cf. [ MS98 ]. Setting Λ := dt then makes H := (Ω , Λ)a stable Hamiltonian structure with Reeb vector field R , and its closed orbits inhomotopy classes that project to S with degree one are in 1-to-1 correspondencewith the 1-periodic Hamiltonian orbits on W . Notice that this is very different fromthe contact case: ξ = ker dt is as far as possible from being a contact structure, itis instead an integrable distribution whose integral submanifolds are the fibers of M → S . Exercise . Show that for any stable Hamiltonian structure H = (Ω , Λ), theflow of R preserves ξ = ker Λ along with its symplectic bundle structure Ω | ξ . Definition . A T -periodic orbit x : R → M of R is called nondegenerate if 1 is not an eigenvalue of dϕ T | ξ x (0) : ξ x (0) → ξ x (0) , where ϕ t denotes the flow of R . Exercise . Show that in Example 6.6, the notions of nondegeneracy forclosed Reeb orbits on M and for 1-periodic Hamiltonian orbits on W (see Lecture 1)coincide.If γ : S → M parametrizes a T -periodic orbit of R with ˙ γ = T · R ( γ ), then theformula of Lecture 3 for the asymptotic operatorA γ η = − J ( ∇ t η − T ∇ η R )still makes sense in this more general context, and it defines an L -symmetric oper-ator on the Hermitian vector bundle ( γ ∗ ξ, J, Ω) over S . It can also be interpretedas a Hessian at a critical point, though for an action functional that is only lo-cally defined: indeed, while Ω need not be globally exact, it is necessarily exact ona neighborhood of γ ( S ) for any given loop γ : S → M , so one can pick anyprimitive λ of Ω on this neighborhood and, for a sufficiently small neighborhood U ( γ ) ⊂ C ∞ ( S , M ) of γ , consider the action functional(6.3) A H : U ( γ ) → R : γ Z S γ ∗ λ. ectures on Symplectic Field Theory Its first variation at γ ∈ U ( γ ) in the direction η ∈ Γ( γ ∗ ξ ) is then d A H ( γ ) η = − Z S Ω( ˙ γ, η ) dt = h− J π ξ ˙ γ, η i L , where π ξ : T M → ξ denotes the projection along R and the L -pairing on γ ∗ ξ is defined via the bundle metric Ω( · , J · ) | ξ . This leads us to interpret − J π ξ ˙ γ as a“gradient” ∇A H ( γ ), and if ˙ γ = T · R ( γ ), then differentiating this gradient in thedirection of η ∈ Γ( γ ∗ ξ ) gives A γ η . As one would expect, nondegeneracy of γ isthen equivalent to the condition ker A γ = { } , and one can in this case define theConley-Zehnder index µ τ CZ ( γ ) ∈ Z as in Lecture 3, relative to a choice of unitarytrivialization τ for ( ξ, J, Ω).
Exercise . In the setting of Example 6.6, work out the relationship between A H and the symplectic action functional for Hamiltonian systems that we discussedin Lecture 1. (Try not to worry too much about signs.) Definition . Given a stable Hamiltonian structure H = (Ω , Λ), denote by J ( H ) ⊂ J ( R × M )the space of smooth almost complex structures J on R × M with the followingproperties: • J is invariant under the R -action on R × M by translation of the first factor; • J ∂ r = R and J R = − ∂ r , where r denotes the natural coordinate on thefirst factor; • J ( ξ ) = ξ and J | ξ is compatible with the symplectic vector bundle structureΩ | ξ .Notice that if H = ( dα, α ) for a contact form α , then J ( H ) matches the space J ( α ) defined in Lecture 1. Exercise . Show that every J ∈ J ( H ) is tamed by all of the symplecticstructures ω ϕ as defined in (6.1) for ϕ ∈ T .Given J ∈ J ( H ), we define the energy of a J -holomorphic curve u : (Σ , j ) → ( R × M, J ) by E ( u ) := sup ϕ ∈T Z Σ u ∗ ω ϕ . Exercise 6.12 above implies that E ( u ) ≥
0, with equality if and only if u is constant.In the contact case, this notion of energy is not identical to the “Hofer energy” thatwe defined in Lecture 1, nor to Hofer’s original definition from [ Hof93 ], but all threeare equivalent for our purposes since uniform bounds on any of them imply uniformbounds on the others.Just as in the contact case, the simplest example of a finite-energy J -holomorphiccurve is a trivial cylinder u γ : R × S → R × M : ( s, t ) ( T s, γ ( t )) , where γ : S → M is a “constant velocity” parametrization of a T -periodic orbit of R , i.e. ˙ γ = T · R ( γ ). More generally, given a punctured Riemann surface ( ˙Σ = Σ \ Γ , j ) Chris Wendl with Γ = Γ + ∪ Γ − , we consider asymptotically cylindrical J -holomorphic curves u : ( ˙Σ , j ) → ( R × M, J ), which are assumed to have the property that for each z ∈ Γ ± , there exist holomorphic cylindrical coordinates identifying a puncturedneighborhood ˙ U z ⊂ ˙Σ of z with Z + = [0 , ∞ ) × S or Z − = ( −∞ , × S respectively,and a trivial cylinder u γ z : R × S → R × M such that u ( s, t ) = exp u γz ( s,t ) h z ( s, t ) for | s | sufficiently large , where h z ( s, t ) is a vector field along u γ z satisfying | h z ( s, · ) | → s →±∞ . As usual, both the norm | h z ( s, t ) | and the exponential map here are assumedto be defined with respect to a translation-invariant choice of Riemannian metricon R × M . The vector fields h z along u γ z for each z ∈ Γ are sometimes called asymptotic representatives of u near z .Asymptotic representatives satisfy a regularity estimate that will be important toknow about, though its proof (given originally in [ HWZ96 ]) would be too lengthy topresent here. The methods behind the following statement involve a combination ofnonlinear regularity arguments as in Lecture 2 with the asymptotic elliptic estimatesfrom Lecture 4. To prepare for the statement, note that H induces a splitting ofcomplex vector bundles(6.4) T ( R × M ) = ǫ ⊕ ξ, where ǫ denotes the trivial complex line bundle generated by the vector field ∂ r , orequivalently, the Reeb vector field. It follows that if γ : S → M is a Reeb orbitand u γ : R × S → R × M is the corresponding trivial cylinder, then any unitarytrivialization τ of the Hermitian bundle ( γ ∗ ξ, J, Ω) naturally induces a trivializationof u ∗ γ T ( R × M ). Proposition
HWZ96 ]) . Assume J ∈ J ( H ) , u : ( ˙Σ , j ) → ( R × M, J ) is J -holomorphic and asymptotically cylindrical, and its asymptotic orbit γ z at z ∈ Γ ± is nondegenerate. Let h ( s, t ) ∈ C n denote the asymptotic representative of u near z expressed via the trivialization induced by a choice of unitary trivializationfor ( γ ∗ z ξ, J, Ω) . If δ > is small enough so that the asymptotic operator A γ z has noeigenvalues in the closed interval between and ∓ δ , then h ( s, t ) = e ∓ δs g ( s, t ) for some bounded function g ( s, t ) ∈ C n whose derivatives of all orders are boundedas s → ±∞ . Remark . The range of δ > δ slightly, one can equivalently say that h ( s, t ) = e ∓ δs g ( s, t ) where thederivatives of all orders of g ( s, t ) decay to zero as s → ±∞ . Exercise . Convince yourself that the analogue of Proposition 6.13 in Morsetheory is true. Namely, suppose (
M, g ) is a Riemannian manifold, f : M → R issmooth and u : R → M is a solution to ˙ u + ∇ f ( u ) = 0 with lim s →±∞ u ( s ) = x ± ∈ Crit( f ), where x ± are nondegenerate critical points. We can write u ( s ) asymptoti-cally as u ( s ) = exp x ± h ± ( s ) ectures on Symplectic Field Theory for some functions h ± ( s ) ∈ T x ± M that are defined for s close to ±∞ and satisfy | h ± ( s ) | → s → ±∞ . Show that if δ > ∇ f ( x ± ) hasno eigenvalue in the closed interval between 0 and ± δ , then h ± ( s ) = e ∓ δs g ± ( s )for some functions g ± ( s ) with bounded derivatives of all orders as s → ±∞ . Hint:fix local coordinates identifying x ± with ∈ R n and first consider the case where ∇ f ( x ) in these coordinates depends linearly on x . Then try to compare u ( s ) withsolutions of this idealized equation. Example . In the setting of Example 6.6, a choice of J ∈ J ( H ) is equiv-alent to a choice of smooth S -parametrized family of compatible almost complexstructures { J t } t ∈ S on ( W, ω ), and J -holomorphic curves u : ( ˙Σ , j ) → ( R × M, J )can be written as u = ( f, v ) : ˙Σ → (cid:0) R × S (cid:1) × W, where f : ( ˙Σ , j ) → ( R × S , i ) is holomorphic. In particular, if ( ˙Σ , j ) = ( R × S , i )and f is taken to have an extension to S → S of degree one, then u can bereparametrized so that f is the identity map, hence u = (Id , v ) : R × S → ( R × S ) × W is a section of the trivial fiber bundle ( R × S ) × W → R × S , and one cancheck that the equation satisfied by v : R × S → W is precisely the Floer equation ∂ s v + J t ( v )( ∂ t v − X t ( v )) = 0 . We discussed symplectic cobordisms between contact manifolds in Lecture 1.Let us now generalize this notion in the context of stable Hamiltonian structures.A symplectic cobordism with stable boundary is a compact symplecticmanifold (
W, ω ) with boundary ∂W = − M − ⊔ M + , equipped with a stabilizingvector field V that points transversely inward at M − and outward at M + . Thisinduces stable Hamiltonian structures H ± = ( ω ± , λ ± ) on M ± , where ω ± := ω | T M ± , λ ± := ( ι V ω ) | T M ± , and observe that the orientation conventions for M + and M − (with the latter car-rying the opposite of the natural boundary orientation) have been chosen such thatif dim W = 2 n , λ ± ∧ ω n − ± > M ± . We can now identify neighborhoods of M ± in ( W, ω ) symplectically with collars ofthe form ([0 , ǫ ) × M + , d ( rλ + ) + ω + ) , (( − ǫ, × M − , d ( rλ − ) + ω − ) , see Figure 6.1. The apparent discrepancy in signs between this and Proposition 6.13 is due to the fact that u ( s ) satisfies a negative gradient flow equation, whereas the nonlinear Cauchy-Riemann equationin symplectizations is interpreted loosely as a positive gradient flow equation. Chris Wendl (( − ǫ, × M + , d ( rλ + ) + ω + )([0 , ǫ ) × M − , d ( rλ − ) + ω − )( W, ω ) Figure 6.1.
A symplectic cobordism with stable boundary compo-nents ∂W = − M − ⊔ M + and symplectic collar neighborhoods inducedby the stable Hamiltonian structures H ± = ( ω ± , λ ± ) on M ± .Modifying (6.2) by(6.5) T := (cid:8) ϕ ∈ C ∞ ( R , ( − ǫ, ǫ )) (cid:12)(cid:12) ϕ ′ > ϕ ( r ) = r for r near 0 (cid:9) , we can use any ϕ ∈ T to define a symplectic completion ( c W , ω ϕ ) of ( W, ω ) by c W := (cid:0) ( −∞ , × M − (cid:1) ∪ M − W ∪ M + (cid:0) [0 , ∞ ) × M + (cid:1) , where the above collar neighborhoods are used to glue the pieces together smoothlyand the symplectic form is defined by ω ϕ := d ( ϕ ( r ) λ − ) + ω − on ( −∞ , × M − ,ω on W ,d ( ϕ ( r ) λ + ) + ω + on [0 , ∞ ) × M + , see Figure 6.2. For each r ≥
0, we define the compact submanifold W r := ([ − r , × M − ) ∪ M − W ∪ M + ([0 , r ] × M + ) , and observe that ( W r , ω ϕ ) is also a symplectic cobordism with stable boundary forevery ϕ ∈ T .Since c W is noncompact, almost complex structures J on c W will need to satisfyconditions near infinity in order for moduli spaces of J -holomorphic curves to be wellbehaved, but we would like to preserve the freedom of choosing arbitrary compatibleor tame almost complex structures in compact subsets. Definition . Given ψ ∈ T and r ≥
0, let J τ ( ω ψ , r , H + , H − ) ⊂ J ( c W )denote the space of smooth almost complex structures J on c W such that: • J on [ r , ∞ ) × M + matches an element of J ( H + ); • J on ( −∞ , − r ] × M − matches an element of J ( H − ); • J on W r is tamed by ω ψ . ectures on Symplectic Field Theory ( W, ω )(( − ǫ, × M + , d ( rλ + ) + ω + )([0 , ǫ ) × M − , d ( rλ − ) + ω − )([0 , ∞ ) × M + , d ( ϕ ( r ) λ + ) + ω + )(( −∞ , × M − , d ( ϕ ( r ) λ − ) + ω − ) Figure 6.2.
The completion ( c W , ω ϕ ) of a symplectic cobordismwith stable boundary.Let J ( ω ψ , r , H + , H − ) ⊂ J τ ( ω ψ , r , H + , H − )denote the subset for which J is additionally compatible with ω ψ on W r .Setting(6.6) T ( ψ, r ) := (cid:8) ϕ ∈ T (cid:12)(cid:12) ϕ ≡ ψ on [ − r , r ] (cid:9) , Exercise 6.12 implies that every J ∈ J ( ω ψ , r , H + , H − ) is tamed by ω ϕ for every ϕ ∈ T ( ψ, r ). It is therefore sensible to define the energy of a J -holomorphic curve u : (Σ , j ) → ( c W , J ) by E ( u ) := sup ϕ ∈T ( ψ,r ) Z Σ u ∗ ω ϕ . The notion of asymptotically cylindrical J -holomorphic curves extends in a straight-forward way to the setting of ( c W , J ): such curves are proper maps whose posi-tive/negative punctures are asymptotic to closed orbits of the Reeb vector field R ± induced by H ± on {±∞} × M ± , see Figure 6.3. The exponential decay estimatein Proposition 6.13 is also immediately applicable in this more general setting sinceasymptotically cylindrical curves in c W are indistinguishable near their puncturesfrom curves in the symplectizations R × M ± .It is easy to check that asymptotically cylindrical J -holomorphic curves alwayshave finite energy. We will prove in Lecture 8 that the converse is also true wheneverthe Reeb orbits are nondegenerate. Chris Wendl c W ˙Σ u Figure 6.3.
An asymptotically cylindrical holomorphic curve in( c W , J ) with genus 2, one positive puncture and two negative punc-tures.
Remark . Strictly speaking, the “trivial stable cobordism”([0 , × M, d ( ϕ ( r )Λ , Ω))induces different stable Hamiltonian structures at M − := { } × M and M + := { } × M , thus one cannot technically regard J ( H ) as contained in any space ofthe form J ( ω ψ , r , H + , H − ) without inventing questionable new notions such asthe “infinitesimal trivial cobordism” [0 , × M (whose completion would be thesymplectization of ( M, H )). It is nonetheless true for fairly trivial reasons that mostresults about J ( ω, r , H + , H − ) apply equally well to J ( H ), and we shall use thisfact in the following without always mentioning it.Every asymptotically cylindrical curve u : ˙Σ → c W has a well-defined relativehomology class , meaning the following. Denote the asymptotic orbits of u atits punctures z ∈ Γ ± by γ z , and let ¯ γ ± ⊂ M ± denote the closed 1-dimensionalsubmanifold defined as the union over z ∈ Γ ± of the images of the orbits γ z . Let Σdenote the compact oriented topological surface with boundary obtained from ˙Σ byappending {±∞} × S to each of its cylindrical ends, and let π : c W → W denotethe retraction defined as the identity on W and π ( r, x ) = x ∈ M ± ⊂ ∂W for ( r, x )in [0 , ∞ ) × M + or ( −∞ , × M − . Then π ◦ u : ˙Σ → W has a natural continuousextension ¯ u : (Σ , ∂ Σ) → ( W, ¯ γ + ∪ ¯ γ − )and thus represents a relative homology class[ u ] ∈ H ( W, ¯ γ + ∪ ¯ γ − ) . ectures on Symplectic Field Theory We continue in the setting of a completed symplectic cobordism c W with fixedchoices of ψ ∈ T , r ≥ J ∈ J ( ω ψ , r , H + , H − ). We shall denote by ξ ± and R ± the hyperplane distribution and Reeb vector field respectively determined bythe stable Hamiltonian structure H ± = ( ω ± , λ ± ).Fix integers g, m, k + , k − ≥ γ ± = ( γ ± , . . . , γ ± k ± ) , where each γ ± i is a closed orbit of R ± in M ± . Denote the union of the images of the γ ± i by ¯ γ ± ⊂ M ± , and choose a relative homology class A ∈ H ( W, ¯ γ + ∪ ¯ γ − )whose image under the boundary map H ( W, ¯ γ + ∪ ¯ γ − ) ∂ −→ H (¯ γ + ∪ ¯ γ − ) definedvia the long exact sequence of the pair ( W, ¯ γ + ∪ ¯ γ − ) is ∂A = k + X i =1 [ γ + i ] − k − X i =1 [ γ − i ] ∈ H (¯ γ + ∪ ¯ γ − ) . The moduli space of unparametrized J -holomorphic curves of genus g with m marked points , homologous to A and asymptotic to ( γ + , γ − ) is then definedas a set of equivalence classes of tuples M g,m ( J, A, γ + , γ − ) = (cid:8) (Σ , j, Γ + , Γ − , Θ , u ) (cid:9) (cid:14) ∼ , where:(1) (Σ , j ) is a closed connected Riemann surface of genus g ;(2) Γ + = ( z +1 , . . . , z + k + ), Γ − = ( z − , . . . , z − k − ) and Θ = ( ζ , . . . , ζ m ) are disjointordered sets of distinct points in Σ;(3) u : ( ˙Σ := Σ \ (Γ + ∪ Γ − ) , j ) → ( c W , J ) is an asymptotically cylindrical J -holomorphic map with [ u ] = A , asymptotic at z ± i ∈ Γ ± to γ ± i for i =1 , . . . , k ± ;(4) Equivalence(Σ , j , Γ +0 , Γ − , Θ , u ) ∼ (Σ , j , Γ +1 , Γ − , Θ , u )means the existence of a biholomorphic map ψ : (Σ , j ) → (Σ , j ), takingΓ ± to Γ ± and Θ to Θ with the ordering preserved, such that u ◦ ψ = u . We shall usually abuse notation by abbreviating elements [(Σ , j, Γ + , Γ − , Θ , u )] inthis moduli space by u ∈ M g,m ( J, A, γ + , γ − ) . The automorphism group
Aut( u ) = Aut(Σ , j, Γ + , Γ − , Θ , u )of u is defined as the group of biholomorphic maps ψ : (Σ , j ) → (Σ , j ) which actas the identity on Γ + ∪ Γ − ∪ Θ and satisfy u = u ◦ ψ . Clearly the isomorphismclass of this group depends only on the equivalence class [(Σ , j, Γ + , Γ − , Θ , u )] ∈ Chris Wendl M g,m ( J, A, γ + , γ − ), and we will see in § u : ˙Σ → c W is constant. The significance of the marked points is that they determinean evaluation map ev : M g,m ( J, A, γ + , γ − ) → c W m : [(Σ , j, Γ + , Γ − , Θ , u )] ( u ( ζ ) , . . . , u ( ζ m ))where Θ = ( ζ , . . . , ζ m ). For most of our applications we will be free to assume m = 0, as marked points are not needed for defining the most basic versions ofSFT; the evaluation map does play a prominent role however in more algebraicallyelaborate versions of the theory, and especially in the Gromov-Witten invariants(the “closed case” of SFT).We will assign a topology to M g,m ( J, A, γ + , γ − ) in the next lecture by locallyidentifying it with subsets of certain manifolds of maps ˙Σ → c W with Sobolev-typeregularity and exponential decay conditions at the ends. In reality, this topologyadmits a simpler description: one can define convergence of a sequence[(Σ ν , j ν , Γ + ν , Γ − ν , Θ ν , u ν )] → [(Σ , j, Γ + , Γ − , Θ , u )]to mean that for sufficiently large ν , the equivalence classes in the sequence admitrepresentatives of the form (Σ , j ′ ν , Γ + , Γ − , Θ , u ′ ν ) such that(1) j ′ ν → j in C ∞ ;(2) u ′ ν → u in C ∞ loc ( ˙Σ , c W );(3) ¯ u ′ ν → ¯ u in C (Σ , W ).The proof that this topology matches what we will define in the next lecture interms of weighted Sobolev spaces requires asymptotic elliptic regularity argumentsalong the lines of Proposition 6.13. In Lecture 2, we proved that closed J -holomorphic curves are all either embeddedin the complement of a finite set or are multiple covers of curves with this property.The same thing holds in the punctured case: Theorem . Assume u : ( ˙Σ , j ) → ( c W , J ) is a nonconstant asymptoti-cally cylindrical J -holomorphic curve whose asymptotic orbits are all nondegenerate,where ˙Σ = Σ \ Γ for some closed Riemann surface (Σ , j ) and finite subset Γ ⊂ Σ .Then there exists a factorization u = v ◦ ϕ , where • ϕ : (Σ , j ) → (Σ ′ , j ′ ) is a holomorphic map of positive degree to anotherclosed and connected Riemann surface (Σ ′ , j ′ ) ; • v : ( ˙Σ ′ , j ′ ) → ( c W , J ) is an asymptotically cylindrical J -holomorphic curvewhich is embedded except at a finite set of critical points and self-intersections,where ˙Σ ′ := Σ ′ \ Γ ′ with Γ ′ := ϕ (Γ) and Γ = ϕ − (Γ ′ ) . As in the closed case, we call u a simple curve if the holomorphic map ϕ :(Σ , j ) → (Σ ′ , j ′ ) is a diffeomorphism, and u is otherwise a k -fold multiple cover of v with k := deg( ϕ ) ≥ ectures on Symplectic Field Theory closed case, our proof required two lemmas which described the local picture of a J -holomorphic curve u : ˙Σ → c W near either a double point u ( z ) = u ( z ) for z = z ora critical point du ( z ) = 0. Both statements were completely local and thus equallyvalid for non-closed curves, but we now need similar statements to describe whatkinds of singularities can appear in the neighborhood of a puncture. The followinglemma is due to Siefring [ Sie08 ] and follows from a “relative asymptotic formula”analogous to Proposition 6.13.
Lemma . Assume u : ( ˙Σ = Σ \ Γ , j ) → ( c W , J ) is asymptot-ically cylindrical and is asymptotic at z ∈ Γ to a nondegenerate Reeb orbit. Thena punctured neighborhood ˙ U z ⊂ ˙Σ of z can be identified biholomorphically with thepunctured disk ˙ D = D \ { } such that u ( z ) = v ( z k ) for z ∈ ˙ D = ˙ U z , where k ∈ N and v : ( ˙ D , i ) → ( c W , J ) is an embedded and asymptotically cylindrical J -holomorphic curve. Moreover, if u ′ : ( ˙Σ ′ = Σ ′ \ Γ ′ , j ′ ) → ( c W , J ) is anotherasymptotically cylindrical curve with a puncture z ′ ∈ Γ ′ , then the images of u near z and u ′ near z ′ are either identical or disjoint. (cid:3) Exercise . With Lemma 6.20 in hand, adapt the proof of Theorem 2.29 inLecture 2 to prove Theorem 6.19. If you get stuck, see [
Nel15 , § Proposition . If [(Σ , j, Γ + , Γ − , Θ , u )] ∈ M g,m ( J, A, γ + , γ − ) is representedby a simple curve, then Aut( u ) is trivial. If it is represented by a k -fold cover of asimple curve, then | Aut( u ) | ≤ k . In particular, Aut( u ) is always finite unless u isconstant. Proof. If u is simple, then it is a diffeomorphism onto its image in a smallneighbourhood of some point, and any map ϕ satisfying u = u ◦ ϕ would be theidentity on such a neighbourhood. By unique continuation, we conclude that Aut( u )is trivial. In general if u = v ◦ ϕ for some simple v : Σ ′ → W and ϕ : Σ → Σ ′ a k -fold branched cover, we haveAut( u ) = { f : Σ → Σ | v ◦ ϕ ◦ f = v ◦ ϕ } . By a similar argument as in the previous case, knowing that v is simple implies weonly need to look at solutions to ϕ ◦ f = ϕ. Remove the set of branch points B from Σ ′ together with the set ϕ − ( B ) fromΣ, so that ϕ becomes an honest covering map. Any ϕ ∈ Aut( u ) then defines abiholomorphic deck transformation of the cover, so it remains to argue that thereare at most k of them. In fact, there is at most one transformation that takes w to w for any two given points w , w ∈ ϕ − ( x ). If there were two such transformations Chris Wendl f and g , then f ◦ g − would be the identity on an open neighbourhood and wouldthus be globally the identity by unique continuation. (cid:3) The following statement, which we will prove in the next lecture, is the maingoal of most of the analysis we have discussed recently. It is essentially an appli-cation of the implicit function theorem for a smooth nonlinear Fredholm sectionof a Banach space bundle. The implicit function theorem (see [
Lan93 ]) implies inparticular that if F is a smooth map between Banach spaces such that F ( x ) = 0and dF ( x ) is a surjective Fredholm operator, then F − (0) is a smooth manifoldnear x with its dimension equal to the Fredholm index of dF ( x ). Surjectivity isan extra hypothesis, referred to in the statement below as “Fredholm regularity,”a notion that we will define precisely in the next lecture. The dimension formulashould look familiar, but is only an indirect consequence of the index formula forCauchy-Riemann type operators that we proved in Lecture 5; one also needs to ac-count for the fact that in defining our moduli space M g,m ( J, A, γ + , γ − ), we did notfix the complex structures on our domain curves, hence they are free to move aboutin the moduli space of Riemann surfaces, whose dimension therefore plays a role indetermining the dimension of M g,m ( J, A, γ + , γ − ). Theorem . The set of
Fredholm regular curves forms an open subset M reg g,m ( J, A, γ + , γ − ) ⊂ M g,m ( J, A, γ + , γ − ) which naturally admits the structure of a smooth finite-dimensional orbifold of di-mension dim M reg g,m ( J, A, γ + , γ − ) = ( n − − g − k + − k − ) + 2 c τ ( A )+ k + X i =1 µ τ CZ ( γ + i ) − k − X i =1 µ τ CZ ( γ − i ) + 2 m, where dim W = 2 n , τ is a choice of unitary trivialization for ( ξ ± , J, ω ± ) along eachof the asymptotic orbits γ ± i , and c τ ( A ) denotes the normal first Chern number of thecomplex vector bundle ( u ∗ T c W , J ) → ˙Σ with respect to the asymptotic trivializationdetermined by τ and the splitting T ( R × M ± ) = ǫ ⊕ ξ ± (cf. (6.4) ). The local isotropygroup of M reg g,m ( J, A, γ + , γ − ) at u is Aut( u ) , hence the moduli space is a manifoldnear any regular element with trivial automorphism group. Exercise . Verify that the number in the above index formula is inde-pendent of the choice of trivializations τ , and that c τ ( u ∗ T c W ) depends only on therelative homology class A .ECTURE 7 Smoothness of the moduli spaceContents C ε to C ∞ In this lecture, we continue the study of the moduli space M ( J ) := M g,m ( J, A, γ + , γ − ) . We assume as before that (
W, ω ) is a 2 n -dimensional symplectic cobordism withstable boundary ∂W = − M − ⊔ M + inheriting stable Hamiltonian structures H ± =( ω ± , λ ± ) with induced Reeb vector fields R ± and hyperplane distributions ξ ± =ker λ ± , g, m, k + , k − ≥ γ ± = ( γ ± , . . . , γ ± k ± ) are ordered sets of peri-odic R ± -orbits in M ± , and A ∈ H ( W, ¯ γ + ∪ ¯ γ − ) is a relative homology class with ∂A = P i [ γ + i ] − P i [ γ − i ] ∈ H ( W, ¯ γ + ∪ ¯ γ − ). The noncompact completion of ( W, ω ) isdenoted by ( c W , ω ψ ) for some fixed function ψ : R → ( − ǫ, ǫ ) that scales the symplec-tic form on the cylindrical ends, and r ≥ r , ∞ ) × M + and ( −∞ , − r ] × M − on which we require our almostcomplex structures J ∈ J ( ω ψ , r , H + , H − ) to be R -invariant. The complement ofthese ends has closure W r := ([ − r , × M − ) ∪ M − W ∪ M + ([0 , r ] × M + ) . We will often make use of the fact that since J matches translation-invariant almostcomplex structures in J ( H ± ) outside of W r , there are natural complex vectorbundle splittings T ( R × M ± ) = ǫ ⊕ ξ ± , where ǫ denotes the canonically trivial line bundle spanned by ∂ r and the Reebvector field. We concluded the previous lecture with the statement of the following theorem.
Chris Wendl
Theorem . If the orbits γ ± i are all nondegenerate and J ∈ J ( ω ψ , r , H + , H − ) ,then the moduli space M ( J ) contains an open subset M reg ( J ) ⊂ M ( J ) consisting of so-called Fredholm regular curves, which naturally admits the structureof a smooth finite-dimensional orbifold of dimension dim M reg ( J ) = ( n − − g − k + − k − ) + 2 c τ ( A )+ k + X i =1 µ τ CZ ( γ + i ) − k − X i =1 µ τ CZ ( γ − i ) + 2 m, where dim W = 2 n , τ is a choice of unitary trivialization for ( ξ ± , J, ω ± ) along eachof the asymptotic orbits γ ± i , and c τ ( A ) denotes the normal first Chern number of thecomplex vector bundle ( u ∗ T c W , J ) → ˙Σ with respect to the asymptotic trivializationdetermined by τ and the splitting T ( R × M ± ) = ǫ ⊕ ξ ± . The local isotropy groupof M reg ( J ) at u is Aut( u ) , hence the moduli space is a manifold near any regularelement with trivial automorphism group. The integer in the above dimension formula is often called the virtual dimen-sion of M ( J ) and denoted byvir-dim M ( J ) := ( n − − g − k + − k − ) + 2 c τ ( A )+ k + X i =1 µ τ CZ ( γ + i ) − k − X i =1 µ τ CZ ( γ − i ) + 2 m. Ignoring the marked points, the virtual dimension of a space M g, ( J, A, γ + , γ − )containing a curve u : ( ˙Σ , j ) → ( c W , J ) with punctures z ∈ Γ ± and nondegenerateasymptotic orbits { γ z } z ∈ Γ ± is sometimes also called the index of u ,ind( u ) := ( n − χ ( ˙Σ) + 2 c τ ( u ∗ T c W ) + X z ∈ Γ + µ τ CZ ( γ z ) − X z ∈ Γ − µ τ CZ ( γ z ) ∈ Z , and we will see that it is in fact the Fredholm index of an operator closely relatedto the linearized Cauchy-Riemann operator D u at u . The word “virtual” refers tothe fact that in general, the regularity condition may fail and thus M ( J ) might notbe smooth, or if it is, it might actually be of a different dimension (see Example 7.5below), but in an ideal world where transversality is always satisfied, its dimensionwould be vir-dim M ( J ). This notion makes sense in finite-dimensional contexts aswell: if f : R n → R m is a smooth map, then we would say that f − (0) has virtualdimension n − m , even though f − (0) might in general be all sorts of strange thingsother than a smooth ( n − m )-dimensional manifold. In particular, n − m couldbe negative, in which case f − (0) would be empty if transversality were satisfied,but in general this need not be the case. It is true however that f can always be perturbed to a map whose zero set is an ( n − m )-dimensional manifold (or emptyif n − m < ectures on Symplectic Field Theory provided by the compatification of M ( J ), which is usually crucial for meangingfulapplications. Such issues require more sophisticated methods than we will discusshere, but a good place to read about them is [ FFGW ].The first goal of this lecture is to define the notion “Fredholm regular” and proveTheorem 7.1. In practice, however, Fredholm regularity is a technical condition thatcan rarely be directly checked. To remedy this, we will also prove a genericity resultfor somewhere injective J -holomorphic curves. A smooth map u : ˙Σ → c W is said tohave an injective point z ∈ ˙Σ if du ( z ) : T z ˙Σ → T u ( z ) c W is injective and u − ( u ( z )) = { z } . If u is a proper map, then it is easy to see that the set of injective points is open in ˙Σ,though in general it could also be empty; this is the case e.g. for multiply covered J -holomorphic curves. We say u is somewhere injective if its set of injective points isnonempty; for asymptotically cylindrical J -holomorphic curves with nondegenerateasymptotic orbits, Theorem 6.19 implies that somewhere injectivity is equivalent tobeing simple , i.e. not multiply covered.Recall that if X is a topological space, a subset Y ⊂ X is called comeager if itcontains a countable intersection of open and dense sets. If X is complete, then theBaire category theorem implies that comeager subsets are always dense; moreover,any countable intersection of comeager subsets is also comeager and therefore dense.Comeager subsets often play the role in infinite dimensions that the term “almosteverywhere” plays in finite dimensions. Informally, we often say that a given state-ment dependent on a choice of auxiliary data (living in a complete metric space)is true generically , or “for generic choices,” if it is true whenever the data arechosen from some comeager subset of the space of all possible data. Theorem . Fix the same data as in Theorem 7.1, an almost complex struc-ture J fix ∈ J ( ω ψ , r , H + , H − ) and an open subset U ⊂ W r . Then there exists a comeager subset J reg U ⊂ n J ∈ J ( ω ψ , r , H + , H − ) (cid:12)(cid:12) J = J fix on c W \ U o , such that for every J ∈ J reg U , every curve u ∈ M ( J ) that has an injective pointmapped into U is Fredholm regular. In particular, the curves with this propertydefine an open subset of M ( J ) that is a smooth manifold with dimension equal toits virtual dimension. Remark . Since
U ⊂ c W has compact closure, the set n J ∈ J ( ω ψ , r , H + , H − ) (cid:12)(cid:12) J = J fix on c W \ U o Elsewhere in the symplectic literature, comeager subsets are sometimes referred to as “setsof second category,” which is unfortunately slightly at odds with the standard meaning of “secondcategory,” though it is accurate to say that the complement of a comeager subset (also known asa “meager” subset) is a set of first category. The term
Baire subset is also sometimes used as asynonym for “comeager subset”. Chris Wendl has a natural C ∞ -topology that makes it a Fr´echet manifold and thus a completemetric space, hence comeager subsets of it are dense. Remark . Both of the above theorems admit easy extensions to the studyof moduli spaces dependent on finitely many parameters. Concretely, suppose P is a smooth finite-dimensional manifold and { J s } s ∈ P is a smooth family of almostcomplex structures satisfying the usual conditions. One can then define a parametricmoduli space M ( { J s } s ∈ P ) = (cid:8) ( s, u ) (cid:12)(cid:12) s ∈ P, u ∈ M ( J s ) (cid:9) and a notion of parametric regularity for pairs ( s, u ) ∈ M ( { J s } ), which is again anopen condition, such that the space M reg ( { J s } ) of parametrically regular elementswill be an orbifold of dimensiondim M reg ( { J s } ) = vir-dim M ( J ) + dim P. Similarly, one can show that if the family { J s } s ∈ P is allowed to vary on an opensubset U ⊂ W r for s lying in some precompact open subset V ⊂ P , then allelements ( s, u ) for which s ∈ V and u has an injective point mapping to U willbe parametrically regular. See [ Wend , § P = [0 ,
1] with V = (0 , generic homotopies of almost complex structures. Here it is important to observe that while regularityin the sense of Theorem 7.1 always implies parametric regularity, the converse isfalse: there can exist parametrically regular pairs ( s, u ) ∈ M ( { J s } ) for which u is not a Fredholm regular element of M ( J s ), hence M ( { J s } ) may be smooth even if M ( J s ) is not smooth for some s ∈ P . This can happen in particular whenever s isa critical value of the projection map M ( { J s } ) → P : ( s, u ) s, see Figure 7.1. In general these cannot be excluded by making generic choices of thehomotopy, though it is possible in certain cases using “automatic” transversality re-sults, which guarantee regularity for all J s with no need for genericity (cf. [ Wen10 ]).
Example . It is not hard to imagine situations in which transversality must fail generically for multiply covered curves. Suppose for instance that (
W, ω ) is an8-dimensional symplectic manifold with compatible almost complex structure J ,and u : S → W is a simple J -holomorphic sphere with no punctures and [ u ] = A ∈ H ( W ), where c ( A ) = −
1. This means u represents an element of a modulispace M , ( J , A ) withvir-dim M , ( J , A ) = 2 − g + 2 c ( A ) = 0 . In particular if u is regular and { J s ∈ J ( ω ) } s ∈ R k is a smooth k -parameter familyof compatible almost complex structures including J , then Remark 7.4 implies thata neighborhood of (0 , u ) in the parametric moduli space M ( { J s } ) = { ( s, u ) | s ∈ P, u ∈ M , ( J s , A ) } is a smooth k -dimensional manifold, and this will be true nomatter how the family { J s } is chosen. But for each of the elements ( s, u ) ∈ M ( { J s } ) ectures on Symplectic Field Theory s M ( J ) M ( J ) Figure 7.1.
The picture shows a smooth parametric moduli space M ( { J s } s ∈ [0 , ) and its projection M ( { J s } ) → [0 ,
1] : ( s, u ) s in acase where vir-dim M ( J s ) = 0. The parametric moduli space is 1-dimensional and the spaces M ( J s ) are regular and 0-dimensional foralmost every s ∈ [0 , s is a criticalvalue of the projection; in the picture, one such space M ( J s ) containsa 1-dimensional component consisting of non-regular curves, so itsdimension differs from its virtual dimension.parametrized by a J -holomorphic map u : ( S = C ∪ {∞} , i ) → ( W, J s ), there isalso a double cover u ′ : S → W : z u ( z ) , with [ u ′ ] = 2 A , so u ′ ∈ M , ( J s , A ) andvir-dim M , ( J s , A ) = 2 − g + 2 c (2 A ) = − . Negative virtual dimension means that M , ( J , A ) should be empty wheneverFredholm regularity is achieved, but this is clearly impossible, even generically, sinceelements of M , ( J s , A ) always have double covers belonging to M , ( J s , A ). Remark . The most common way to apply Theorem 7.2 is by setting U equal to the interior of W r , so generic perturbations of J are allowed everywhereexcept on the regions where it is required to be R -invariant. The theorem thenachieves transversality for all simple curves that are not confined to the R -invariantregions. We will show in the next lecture that transversality for all curves of thelatter type can also be achieved by generic perturbations within the spaces J ( H ± ) ofcompatible R -invariant almost complex structures on the symplectizations R × M ± ,hence generic choices in J ( ω ψ , r , H + , H − ) do achieve transversality for all simplecurves.Our proofs of Theorems 7.1 and 7.2 will mostly follow the same line of argumentthat is carried out for the closed case in [ Wend , Chapter 4], thus we will not discussevery detail but will instead emphasize aspects which are unique to the puncturedcase. Chris Wendl
Fix k ∈ N and p ∈ (1 , ∞ ) with kp >
2, a small number δ ≥
0, and a Riemannianmetric on c W that is translation-invariant in the cylindrical ends. Fix also a closedconnected surface Σ of genus g , and disjoint finite ordered sets of distinct pointsΓ ± = ( z ± , . . . , z ± k ± ) , Θ = ( ζ , . . . , ζ m )in Σ, together with disjoint neighborhoods U ± j ⊂ Σof each z ± j ∈ Γ ± with complex structures j Γ and biholomorphic identifications of( U ± j , j Γ , z j ) with ( D , i,
0) for each j = 1 , . . . , k ± . This determines holomorphic cylin-drical coordinates identifying each of the punctured neighborhoods˙ U ± j ⊂ ˙Σ := Σ \ (Γ + ∪ Γ − )biholomorphically with the half-cylinder Z ± .For reasons that will become clear when we study the linearized Cauchy-Riemannoperator in the punctured setting, we will need to consider exponentially weightedSobolev spaces. Suppose E → ˙Σ is an asymptotically Hermitian vector bundle:then the Banach space W k,p,δ ( E ) ⊂ W k,p loc ( E )is defined to consist of sections η ∈ W k,p loc ( E ) whose representatives f : Z ± → C m incylindrical coordinates ( s, t ) ∈ Z ± and asymptotic trivializations at the ends satisfy(7.1) k e ± δs f k W k,p ( Z ± ) < ∞ . The norm of a section η ∈ W k,p,δ ( E ) is defined by adding the W k,p -norm of η over alarge compact subdomain in ˙Σ to the weighted norms (7.1) for each cylindrical end.If δ = 0, this just produces the usual W k,p ( E ), but for δ >
0, sections in W k,p,δ ( E )are guaranteed to have exponential decay at infinity. Remark . It is occasionally useful to observe that the definition of W k,p,δ ( E )also makes sense when δ <
0. In this case, sections in W k,p,δ ( E ) are of class W k,p loc but need not be globally in W k,p ( E ), as they are also allowed to have exponential growth at infinity.We now want to define a Banach manifold of maps u : ˙Σ → c W that will containall the asymptotically cylindrical J -holomorphic curves with our particular choiceof asymptotic orbits. Recall that the asymptotically cylindrical condition means(7.2) u ( s, t ) = exp ( T ± j s,γ ± j ( t )) h ( s, t ) for sufficiently large | s | in suitable cylindrical coordinates ( s, t ) ∈ Z ± near each puncture z ± j ∈ Γ ± , where T ± j > γ ± j : S → M ± and h ( s, t ) is a vector field along thetrivial cylinder that decays as s → ±∞ . The catch is that this definition was notformulated with respect to a fixed choice of the holomorphic cylindrical coordinates( s, t ); in general the coordinates in which (7.2) is valid may depend on u , and differentchoices of coordinates might be required for different maps. One can show however ectures on Symplectic Field Theory that any two distinct choices of holomorphic cylindrical coordinates are related toeach other by a transformation that converges asymptotically to a constant shift,which implies that for our fixed choice of coordinates ( s, t ), every asymptoticallycylindrical map can be assumed to satisfy u ( s, t ) = exp ( T ± j s + a,γ ± j ( t + b )) h ( s, t ) , lim s →±∞ h ( s, t ) = 0for some constants a ∈ R and b ∈ S . We therefore define the space B k,p,δ := W k,p,δ ( ˙Σ , c W ; γ + , γ − ) ⊂ C ( ˙Σ , c W )to consist of all continuous maps u : ˙Σ → c W of the form u = exp f h, where: • f : ˙Σ → c W is smooth and, in our fixed cylindrical coordinates ( s, t ) ∈ Z ± on neighborhoods of the punctures z ± j ∈ Γ ± , takes the form f ( s, t ) = ( T ± j s + a, γ ± j ( t + b )) for | s | sufficiently large,where a ∈ R and b ∈ S are arbitrary constants and T ± j > γ ± j : S → M ± ; • h ∈ W k,p,δ ( f ∗ T c W ).Though it is not immediate since ˙Σ is noncompact, one can generalize the ideasin [ El˘ı67 ] to give B k,p,δ the structure of a smooth, separable and metrizable Banachmanifold. The key point is the condition kp >
2, which guarantees the continuousinclusion W k,p,δ ( f ∗ T c W ) ֒ → C ( f ∗ T c W ) as well as Banach algebra and C k -continuityproperties, cf. Propositions 2.4, 2.7 and 2.8 in Lecture 2. These properties areneeded in order to show that the transition maps between pairs of charts of theform exp f h h are smooth.The tangent space to B k,p,δ at u ∈ B k,p,δ can be written as T u B k,p,δ = W k,p,δ ( u ∗ T c W ) ⊕ V Γ , where V Γ ⊂ Γ( u ∗ T c W ) is a non-canonical choice of a 2( k + + k − )-dimensional vectorspace of smooth sections asymptotic at the punctures to constant linear combina-tions of the vector fields spanning the canonical trivialization of the first factor in T ( R × M ± ) = ǫ ⊕ ξ ± , i.e. they point in the R - and R ± -directions. The space V Γ appears due to the fact that two distinct elements of B k,p,δ are generally asymptoticto collections of trivial cylinders that differ from each other by k + + k − pairs ofconstant shifts ( a, b ) ∈ R × S .Fix J ∈ J ( ω ψ , r , H + , H − ) and a smooth complex structure j on Σ that matches j Γ in the neighborhoods U ± j of the punctures. The nonlinear Cauchy-Riemann op-erator is then defined as a smooth section¯ ∂ j,J : B k,p,δ → E k − ,p,δ : u T u + J ◦ T u ◦ j of a Banach space bundle E k − ,p,δ → B k,p,δ Chris Wendl with fibers E k − ,p,δu = W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )) . The zero set of ¯ ∂ j,J is the set of all maps u ∈ B k,p,δ that are pseudoholomorphicfrom ( ˙Σ , j ) to ( c W , J ). Note that the smoothness of ¯ ∂ j,J depends mainly on the factthat J is smooth. Indeed, in local coordinates ¯ ∂ j,J looks like u ∂ s u + ( J ◦ u ) ∂ t u ,in which the most obviously nonlinear ingredient is u J ◦ u . If J were only ofclass C k , then the C k -continuity property would imply that the map u J ◦ u sends maps of class W k,p continuously to maps of class W k,p , and one can use aninductive argument to show that this map then becomes r -times differentiable if J is of class C k + r , see [ Wend , Lemma 2.12.5]. Moreover, the fact that ¯ ∂ j,J u satisfiesthe same exponential weighting condition as u at the cylindrical ends depends onthe fact that J is R -invariant near infinity.For u ∈ ¯ ∂ − j,J (0), the linearization D ¯ ∂ j,J ( u ) : T u B k,p,δ → E k − ,p,δu defines a boundedlinear operator D u : W k,p,δ ( u ∗ T c W ) ⊕ V Γ → W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )) . We derived a formula for this operator in Lecture 2 and showed that it is of Cauchy-Riemann type. Since V Γ is finite dimensional, D u will be Fredholm if and only if itsrestriction to the first factor is Fredholm; denote this restriction by D δ : W k,p,δ ( u ∗ T c W ) → W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )) , where we’ve chosen the notation to emphasize the dependence of this operator onthe choice of exponential weight δ ≥ D δ is Fredholm, consider first the special case where u is a trivialcylinder u γ : R × S → R × M : ( s, t ) ( T s, γ ( t ))over some Reeb orbit γ : S → M with period T > M with stable Hamiltonianstructure H = ( ω, λ ) on M . In this case, there is a more convenient way to writedown D u γ than the formula from Lecture 2. To start with, we use the splitting T ( R × M ) = ǫ ⊕ ξ to decompose u ∗ γ T ( R × M ) = u ∗ γ ǫ ⊕ u ∗ γ ξ and thus write D u γ inblock form D u γ = (cid:18) D ǫu γ D ǫξu γ D ξǫu γ D ξu γ (cid:19) . Exercise . Suppose D : Γ( E ) → Ω , ( ˙Σ , E ) is a linear Cauchy-Riemanntype operator on a vector bundle E with a complex-linear splitting E = E ⊕ E ,and D = (cid:18) D D D D (cid:19) is the resulting block decomposition of D . Use the Leibniz rule satisfied by D to show that D and D are also Cauchy-Riemann type operators on E and E respectively, while the off-diagonal terms are tensorial, i.e. they commute withmultiplication by smooth real-valued functions and thus define bundle maps D : E → Λ , T ∗ ˙Σ ⊗ E and D : E → Λ , T ∗ ˙Σ ⊗ E . ectures on Symplectic Field Theory Now observe that if u = ( u R , u M ) : R × S → R × M is another cylinder near u γ ,the nonlinear operator ( ¯ ∂ j,J u ) ∂ s = ∂ s u + J ∂ t u ∈ Γ( u ∗ T ( R × M )) = Γ( u ∗ ǫ ⊕ u ∗ ξ )takes the form ( ¯ ∂ j,J u ) ∂ s = (cid:18) ∂ s u R − λ ( ∂ t u M ) + i ( ∂ t u R + λ ( ∂ t u M )) π ξ ∂ s u M + J π ξ ∂ t u M (cid:19) , where we are using the canonical trivialization of u ∗ ǫ via ∂ r and R to express the topblock as a complex-valued function. As we observed in Lecture 3, the bottom block ofthis expression can be interpreted in terms of the gradient flow of an action functional A H : C ∞ ( S ) → R , with ∇A H ( γ ) = − J π ξ ∂ t γ . Linearizing in the direction of asection η ξ ∈ Γ( u ∗ γ ξ ) and taking the ξ component thus yields an expression involvingthe Hessian of A H at the critical point γ , namely( D ξu γ η ξ ) ∂ s = ( ∂ s − A γ ) η ξ . To compute the blocks D ǫu γ and D ξǫu γ , notice that D u γ η ǫ = 0 whenever η ǫ is a constantlinear combination of ∂ r and R , as η ǫ is then the derivative of a smooth family of J -holomorphic reparametrizations of u γ . This is enough to prove D ξǫu γ = 0 sincethe latter is tensorial by Exercise 7.8, and expressing arbitrary sections of u ∗ γ ǫ as f ∂ r + gR , we can apply the Leibniz rule for D ǫu γ and conclude( D ǫu γ η ǫ ) ∂ s = ( ∂ s + i ∂ t ) η ǫ in the canonical trivialization. To compute the remaining off-diagonal term, oneneeds to compute dr ( D u γ η ξ ) and λ ( D u γ η ξ ) for an arbitrary section η ξ ∈ Γ( u ∗ γ ξ ),e.g. by picking a smooth family u ρ : R × S → R × M with ∂ ρ u ρ | ρ =0 = η ξ and aconnection ∇ and computing dr (cid:0) ∇ ρ ( ¯ ∂ j,J u ρ ) (cid:12)(cid:12) ρ =0 (cid:1) and λ (cid:0) ∇ ρ ( ¯ ∂ j,J u ρ ) (cid:12)(cid:12) ρ =0 (cid:1) . This calculation is straightforward but unenlightening, so I will leave it as an exercisefor now—in the next lecture we’ll derive a general formula (see Lemma 8.10), whichimplies that since π ξ ∂ s u γ ≡ π ξ ∂ t u γ ≡ D ǫξu γ = 0. All thisleads to the formula ( D u γ η ) ∂ s = (cid:18) ∂ s − (cid:18) − i∂ t A γ (cid:19)(cid:19) η. Here the upper left block is the “trivial” asymptotic operator acting on the trivialline bundle over S . Since every asymptotically cylindrical curve approximates atrivial cylinder near infinity, one can deduce from this calculuation the following: Proposition . The Cauchy-Riemann type operator D u on u ∗ T c W is as-ymptotic at its punctures z ± j ∈ Γ ± for j = 1 , . . . , k ± to the asymptotic operators ( − i∂ t ) ⊕ A γ ± j on ( γ ± j ) ∗ ( ǫ ⊕ ξ ± ) . Perhaps you can now see a problem: even if the orbits γ ± j are all nondegenerate,the asymptotic operators ( − i∂ t ) ⊕ A γ are degenerate, as they have nontrivial kernelconsisting of constant sections in the first (trivial) factor of ( γ ± j ) ∗ ( ǫ ⊕ ξ ± ). Thisimplies in particular that D : W k,p ( u ∗ T c W ) → W k − ,p (Hom C ( T ˙Σ , u ∗ T c W )) Chris Wendl is not Fredholm, except of course in the special case where there are no punctures.The situation is saved by the exponential weight:
Lemma . For every δ > sufficiently small, the operator D δ is Fredholmand has index ind( D δ ) = nχ (Σ) − ( n + 1) c τ ( u ∗ T c W ) + k + X j =1 µ τ CZ ( γ + j ) − k − X j =1 µ τ CZ ( γ − j ) . Moreover, every element of M ( J ) can be represented by a map u ∈ B k,p,δ . Proof.
The second claim follows from the exponential decay estimate of Hofer-Wysocki-Zehnder [
HWZ96 ] mentioned in the previous lecture, see Proposition 6.13.To see that D δ : W k,p,δ → W k − ,p,δ is Fredholm and to compute its index, wecan identify it with a Cauchy-Riemann type operator from W k,p to W k − ,p . Indeed,pick any smooth function f : ˙Σ → R with f ( s, t ) = ∓ δs on the cylindrical endsnear Γ ± , define Banach space isomorphismsΦ δ : W k,p → W k,p,δ : η e f η, Ψ δ : W k − ,p → W k − ,p,δ : θ e f θ, and consider the bounded linear map D ′ δ := Ψ − δ D δ Φ δ : W k,p ( u ∗ T c W ) → W k − ,p (Hom C ( T ˙Σ , u ∗ T c W )) . Using the Leibniz rule for D δ , it is straightforward to show that D ′ δ is also a linearCauchy-Riemann type operator. Moreover, suppose D δ takes the form ¯ ∂ + S ( s, t ) incoordinates and trivialization on the cylindrical end near z ± j , where S ( s, t ) → S ∞ ( t )as s → ±∞ and A γ ± j = − i∂ t − S ∞ ( t ). Then D ′ δ on this same end takes the form D ′ δ η = e ± δs ( ¯ ∂ + S ( s, t ))( e ∓ δs η ) = ¯ ∂η + ( S ( s, t ) ∓ δ ) η and is therefore asymptotic to the perturbed asymptotic operator˜ A ± j := (cid:16) ( − i∂ t ) ⊕ A γ ± j (cid:17) ± δ. The latter is the direct sum of two asymptotic operators − i∂ t ± δ on the trivialline bundle and A γ ± j ± δ on ( γ ± j ) ∗ ξ ± respectively. Since γ ± j is nondegenerate byassumption and the spectrum of A γ ± j is discrete, we can assume ker( A γ ± j ± δ ) remainstrivial if δ > − i∂ t consists of the integer multiples of 2 π , thus − i∂ t ± δ also becomesnondegenerate for any δ > − i∂ t has a 2-dimensional nullspace consisting of sections with winding number 0, andthis becomes an eigenspace for the smallest positive eigenvalue if the puncture ispositive or the largest negative eigenvalue if the puncture is negative. Theorem 3.36thus gives µ CZ ( − i∂ t ± δ ) = ∓ , ectures on Symplectic Field Theory and therefore, µ τ CZ ( ˜ A ± j ) = ∓ µ τ CZ ( γ ± j ) . Plugging this into the general index formula from Lecture 5 then gives the statedresult. (cid:3)
Putting back the missing 2( D u , we have: Corollary . For all δ > sufficiently small, the linearized Cauchy-Riemannoperator D u : T u B k,p,δ → E k − ,p,δu is Fredholm with index ind( D u ) = nχ (Σ) − ( n − c τ ( u ∗ T c W ) + k + X j =1 µ τ CZ ( γ + j ) − k − X j =1 µ τ CZ ( γ − j ) . Since the moduli space M ( J ) is not defined with reference to any fixed complexstructure on the domains ˙Σ, we must build this freedom into the setup. For a moredetailed version of the following discussion, see [ Wend , § g, ℓ ≥
0, the moduli space of Riemann surfaces of genus g with ℓ marked points is a space of equivalence classes M g,ℓ = { (Σ , j, Θ) } (cid:14) ∼ where (Σ , j ) is a compact connected surface with genus g , Θ ⊂ Σ is an orderedset of ℓ points and equivalence is defined via biholomorphic maps that preservethe marked points with their ordering. This space has been studied extensively inalgebraic geometry, though it can also be understood using the same global analyticmethods that we have been applying for M ( J ). It is known in particular that M g,ℓ is always a smooth orbifold, and for any [(Σ , j, Θ)] ∈ M g,ℓ , it satisfies(7.3) dim Aut(Σ , j, Θ) − dim M g,ℓ = 3 χ (Σ) − ℓ, where Aut(Σ , j, Θ) is the group of biholomorphic transformations of (Σ , j ) thatfix the points in Θ. This group is finite whenever (Σ , j,
Θ) is stable , meaning χ (Σ \ Θ) <
0, and in that case (7.3) turns into the well-known dimension formuladim M g,ℓ = − χ (Σ) + 2 ℓ = 6 g − ℓ. This is also the dimension of the
Teichm¨uller space T (Σ , Θ) := J (Σ) / Diff (Σ , Θ) , where J (Σ) denotes the space of all smooth complex structures on Σ compatiblewith its orientation, and Diff (Σ , Θ) is the identity component of the group of dif-feomorphisms that fix Θ. It is a classical result that T (Σ , Θ) is a smooth manifold ofthe same dimension as M g,ℓ , and indeed, the latter can be presented as the quotientof the former by the discrete action of the mapping class group of (Σ , Θ).Equation (7.3) is actually a formula for a Fredholm index. To see how this works,consider first the case ℓ = 0. The right hand side is then χ (Σ) + 2 c ( T Σ), whichis, according to Riemann-Roch, the index of the natural Cauchy-Riemann operatoron T Σ that defines its holomorphic structure. This operator can also be interpreted Chris Wendl as the linearization at the identity map of the nonlinear
Cauchy-Riemann opera-tor for holomorphic maps (Σ , j ) → (Σ , j ), so its kernel is naturally isomorphic to T Id Aut(Σ , j ). Similarly, one can show that the cokernel of this operator is naturallyisomorphic to T [ j ] T (Σ). This discussion remains valid if marked points are included:the main difference is then that the Cauchy-Riemann operator on T Σ should berestricted to a space of vector fields that vanish at Θ, defining a 2 ℓ -codimensionalsubspace as the domain and thus reducing the index by 2 ℓ .For a proof of the following, see [ Wend , Chapter 4] and [
Wen10 , § Proposition . Given a closed Riemann surface (Σ , j ) with a finite orderedset Θ ⊂ Σ , there exists a smooth finite-dimensional submanifold T ⊂ J (Σ) with thefollowing properties:(1) The map
T → T (Σ , Θ) : j ′ [ j ′ ] is bijective onto a neighborhood of [ j ] in T (Σ , Θ) ;(2) The subspace T j T ⊂
Γ(End C ( T Σ)) is complementary in W k − ,p (End C ( T Σ)) to the image of the standard Cauchy-Riemann operator of T Σ acting on thedomain { X ∈ W k,p ( T Σ) | X | Θ = 0 } ;(3) Every j ′ ∈ T equals j near Θ and is invariant under the action of Aut(Σ , j, Θ) by diffeomorphisms on Σ . (cid:3) We will refer to the family
T ⊂ J (Σ) in this proposition as a
Teichm¨uller slicethrough j . We are now in a position to define the necessary regularity condition and provethat a neighborhood of any given regular element [(Σ , j , Γ + , Γ − , Θ , u )] in M ( J ) isan orbifold of the stated dimension. After reparametrizing, we can assume withoutloss of generality that Σ, Γ ± and Θ are precisely the data that were fixed in § j ∈ J (Σ) matches j Γ on our fixed coordinate neighborhoods of Γ ± . We canthen choose a Teichm¨uller slice T ⊂ J (Σ)through j as provided by Prop. 7.12, but with j in that statement replaced by j and Θ replaced by Γ + ∪ Γ − ∪ Θ. In particular, T is invariant under the action ofthe group G := Aut(Σ , j , Γ + ∪ Γ − ∪ Θ) , and (7.3) now becomes(7.4) dim G − dim T = 3 χ (Σ) − k + + k − + m ) . There is a natural extension of the nonlinear operator ¯ ∂ j,J in § ∂ J : T × B k,p,δ → E k − ,p,δ : ( j, u ) T u + J ◦ T u ◦ j of a Banach space bundle E k − ,p,δ → T × B k,p,δ with fibers E k − ,p,δ ( j,u ) = W k − ,p,δ (cid:0) Hom C (( T ˙Σ , j ) , ( u ∗ T c W , J )) (cid:1) . ectures on Symplectic Field Theory The zero set ¯ ∂ − J (0) ⊂ T × B k,p,δ consists of pairs ( j, u ) for which u : ( ˙Σ , j ) → ( c W , J )is pseudoholomorphic, and it contains ( j , u ) by construction. It also admits anatural action of the automorphism group G , G × ¯ ∂ − J (0) → ¯ ∂ − J (0) : ( ϕ, ( j, u )) ( ϕ ∗ j, u ◦ ϕ ) , whose stabilizer at ( j , u ) is Aut( u ), a finite group whenever u is not constant.Observe that any two elements in the same G -orbit of ¯ ∂ − J (0) define equivalentelements of the moduli space M ( J ), as they are related to each other by a biholo-morphic reparametrization that fixes the punctures and marked points. Lemma . The map ¯ ∂ − J (0) (cid:14) G → M ( J ) : [( j, u )] [(Σ , j, Γ + , Γ − , Θ , u )] is a homeomorphism between open neighborhoods of [( j , u )] and [(Σ , j , Γ + , Γ − , Θ , u )] . Proof.
This depends fundamentally on the same fact underlying the smooth-ness of Teichm¨uller space: the action of Diff (Σ , Γ + ∪ Γ − ∪ Θ) on J (Σ) is free andproper. See the proof of [
Wend , Theorem 4.3.6]. (cid:3)
Definition . We say that [(Σ , j , Γ + , Γ − , Θ , u )] is Fredholm regular ifthere exists a choice of Teichm¨uller slice T through j such that the linearization D ¯ ∂ J ( j , u ) : T j T ⊕ T u B k,p,δ → E k − ,p,δ ( j ,u ) is surjective.One can show that the surjectivity condition in this definition does not actuallydepend on the choice of Teichm¨uller slice. This follows from the identification of T j T with the cokernel of the natural Cauchy-Riemann operator on T ˙Σ; see [ Wend ,Lemma 4.3.2].
Proof of Theorem 7.1.
The fact that M ( J ) is an orbifold in a neighbor-hood of [(Σ , j , Γ + , Γ − , Θ , u )] with isotropy group Aut( u ) follows from Lemma 7.13and the implicit function theorem, which gives ¯ ∂ − J (0) the structure of a finite-dimensional manifold near ( j , u ) if Fredholm regularity is satisfied. There is a bitof work to be done in showing that transition maps relating any two overlappingcharts that arise in this way from the implicit function theorem are smooth; for this,we refer again to the proof of Theorem 4.3.6 in [ Wend ] and merely comment thatthe key ingredient is elliptic regularity.The dimension of M ( J ) isdim M ( J ) = dim ¯ ∂ − J (0) − dim G = ind D ¯ ∂ J ( j , u ) − dim G . The restriction of D ¯ ∂ J ( j , u ) to T u B k,p,δ is the operator D u that we studied in § D ¯ ∂ J ( j , u ) = dim T + ind D u . This is true at least in the stable case, i.e. when χ ( ˙Σ \ Θ) <
0. There are finitely many casesnot satisfying this hypothesis, for which the lemma can be proved by more direct arguments sinceexplicit descriptions of both Teichm¨uller space and the automorphism groups of Riemann surfacesare available; see [
Wen10 , § § Chris Wendl
Using (7.4) to replace dim
T − dim G and combining this with Corollary 7.11 nowgives the stated formula for dim M ( J ). (cid:3) The remainder of this lecture is devoted to the proof of Theorem 7.2. The maintool for this purpose is the Sard-Smale theorem [
Sma65 ], an infinite-dimensionalversion of Sard’s theorem stating that the regular values of a smooth nonlinearFredholm map between separable Banach spaces (i.e. a smooth map whose deriv-ative at every point is a Fredholm operator) form a comeager subset of the targetspace. In order to incorporate perturbations of the almost complex structure intoour functional analytic setup, we need to choose a suitable Banach manifold of al-most complex structures. All known ways of doing this are in some sense non-ideal,e.g. one could take almost complex structures of class C k or W k,p , but this neces-sarily introduces non-smooth almost complex structures into the picture, with theconsequence that the nonlinear Cauchy-Riemann operator has only finitely manyderivatives. That is not the end of the world, and indeed, this is the approach takenin [ MS04 ], but I will instead present an approach that was introduced by Floer in[
Flo88b ], in terms of what is now called the “Floer C ε space”. The idea is to workwith a Banach manifold that continuously embeds into the space of smooth almostcomplex structures, so that the nonlinear Cauchy-Riemann operator will always besmooth. It’s a nice trick, but the catch is that we obtain a space that is strictlysmaller than the actual space of smooth almost complex structures we’re interestedin, and has a much stronger topology. The C ε space should be viewed as a usefultool but not a deeply meaningful object—you might notice that while some of theintermediate results stated below depend on its (somewhat ad hoc) definition, The-orem 7.2 does not. This is due to a general trick described in § C ε into results about C ∞ .As in the statement of Theorem 7.2, assume U ⊂ W r is open and J fix ∈J ( ω ψ , r , H + , H − ). Let J U := n J ∈ J ( ω ψ , r , H + , H − ) (cid:12)(cid:12)(cid:12) J = J fix on c W \ U o , and choose any almost complex structure J ref ∈ J U . We can regard J U as a smooth Fr´echet manifold with tangent spaces T J ref J U = n Y ∈ Γ (cid:0) End C ( T c W , J ref ) (cid:1) (cid:12)(cid:12)(cid:12) Y | c W \U ≡ ω ψ ( · , Y · ) + ω ψ ( Y · , · ) ≡ o , where the antilinearity of Y ∈ T J ref J U means that Y is tangent to the space almostcomplex structures, and the condition relating it to ω ψ means that these structuresare compatible with ω ψ . One can check that the map Y J Y := (cid:18) + 12 J ref Y (cid:19) J ref (cid:18) + 12 J ref Y (cid:19) − ectures on Symplectic Field Theory maps a neighborhood of 0 ∈ T J ref J U bijectively to a neighborhood of J ref in J U .We thus fix a sufficiently small constant c > C ε -smallperturbations of J ref ” by J ε U := ( J Y ∈ J U (cid:12)(cid:12)(cid:12)(cid:12) Y ∈ T J ref J U with ∞ X ℓ =0 ε ℓ k Y k C ℓ ( U ) < c ) , where ε := ( ε ℓ ) ∞ ℓ =0 is a fixed sequence of positive numbers with ε ℓ → ℓ → ∞ .The sum k Y k C ε := ∞ X ℓ =0 ε ℓ k Y k C ℓ ( U ) defines a norm, and the space of smooth sections Y ∈ T J ref J U for which this norm isfinite is then a separable Banach space; see Appendix B for a proof of this statement.This makes J ε U a separable and metrizable Banach manifold, as the map J Y Y can be viewed as a chart identifying it with an open subset of the aforementionedBanach space. Not every J ∈ J U near J ref belongs to J ε U , but there is a continuousinclusion J ε U ֒ → J U , where the latter carries its usual C ∞ -topology and J ε U carries the topology inducedby the C ε -norm. By a lemma due to Floer, choosing a sequence ε ℓ that decayssufficiently fast makes J ε U large enough to contain perturbations in arbitrary direc-tions with arbitrarily small support near arbitrary points in U ; see Theorem B.6 inAppendix B for a precise version of this statement and its proof. We will assumefrom now on that a suitably fast decaying sequence has been fixed.We now define a universal moduli space M ∗ ( J ε U ) := (cid:8) ( u, J ) (cid:12)(cid:12) J ∈ J ε U , u ∈ M ( J ) and u has an injective point mapped into U (cid:9) . The terminology is somewhat unfortunate, as M ∗ ( J ε U ) depends on many auxiliarychoices such as J ref and ( ε ℓ ) ∞ ℓ =0 and thus should not really be thought of as a “uni-versal” object. Nonetheless: Lemma . The universal moduli space M ∗ ( J ε U ) is a smooth separable Banachmanifold, and the projection M ∗ ( J ε U ) → J ε U : ( u, J ) J is smooth. Proof.
As in the proof of Theorem 7.1, one can identify M ∗ ( J ε U ) locally withthe zero set of a smooth section of a Banach space bundle. Suppose J ∈ J ε U and[(Σ , j , Γ + , Γ − , Θ , u )] ∈ M ( J ) where u : ˙Σ → c W has an injective point z with u ( z ) ∈ U . Choose a Teichm¨uller slice T through j as in Proposition 7.12 andconsider the smooth section¯ ∂ : T × B k,p,δ × J ε U → E k − ,p,δ : ( j, u, J ) T u + J ◦ T u ◦ j, where E k − ,p,δ is the obvious extension of our previous Banach space bundle to abundle over T × B k,p,δ × J ε U . We’re assuming as before that k ∈ N , 1 < p < ∞ , Chris Wendl kp >
2, and δ > u , J ) in M ∗ ( J ε U ) can then beidentified with a neighborhood of [( j , u , J )] in¯ ∂ − (0) (cid:14) G , where G := Aut(Σ , j , Γ + ∪ Γ − ∪ Θ) acts on ¯ ∂ − (0) by ϕ · ( j, u, J ) := ( ϕ ∗ j, u ◦ ϕ, J ).Since u has an injective point, Aut( u ) is trivial and the G -action at ( j , u , J )is therefore free; hence it suffices to show that ¯ ∂ − (0) is a smooth Banach manifoldnear ( j , u , J ). This follows from the implicit function theorem if we can show that D ¯ ∂ ( j , u , J ) : T j T ⊕ T u B k,p,δ ⊕ T J J ε U → E k − ,p,δ ( j ,u ,J ) is surjective; indeed, the infinite-dimensional implicit function theorem (see [ Lan93 ])requires the additional hypothesis that D ¯ ∂ ( j , u , J ) has a bounded right inverse,but this is immediate since the restriction of this operator to the factor T u B k,p,δ isFredholm (see Exercise 7.17 below). We claim in fact that T u B k,p,δ ⊕ T J J ε U → E k − ,p,δ ( j ,u ,J ) ( η, Y ) D ¯ ∂ ( j , u , J )(0 , η, Y ) = D u η + Y ◦ T u ◦ j is surjective. Consider first the case k = 1, so we are looking at a bounded linearmap W ,p,δ ( u ∗ T c W ) ⊕ V Γ ⊕ T J J ε U → L p,δ (Hom C ( T ˙Σ , u ∗ T c W )) . Note that the dual of any space of sections of class L p,δ can be identified withsections of class L q, − δ for p + q = 1 (recall Remark 7.7). Indeed, choosing a suitable L -pairing defines a bounded bilinear map(7.5) h , i L : L p,δ × L q, − δ → R , and one can use isomorphisms of the form L p → L p,δ : η e f η as in the proofof Lemma 7.10 to prove ( L p,δ ) ∗ ∼ = L q, − δ as a corollary of the standard fact that( L p ) ∗ ∼ = L q . With this understood, observe that since D u : W ,p,δ ⊕ V Γ → L p,δ is Fredholm, we know by Exercise 7.16 below that the map under considerationhas closed range. Thus if it is not surjective, the Hahn-Banach theorem providesa nontrivial element θ ∈ L q, − δ (Hom C ( T ˙Σ , u ∗ T c W )) that annihilates its image underthe pairing (7.5), which amounts to the two conditions h D u η, θ i L = 0 for all η ∈ W ,p,δ ( u ∗ T c W ) ⊕ V Γ , h Y ◦ T u ◦ j , θ i L = 0 for all Y ∈ T J J ε U . (7.6)The first relation is valid in particular for all smooth sections η with compact supportand thus means that θ is a weak solution to the formal adjoint equation D ∗ u θ = 0;applying elliptic regularity and the similarity principle, θ is therefore smooth and hasonly isolated zeroes. We will see however that this contradicts the second relationas long as there exists an injective point z ∈ ˙Σ with u ( z ) ∈ U . Indeed, since theset of injective points with this property is open and zeroes of θ are isolated, letus assume without loss of generality that θ ( z ) = 0. Then by a standard lemmain symplectic linear algebra (see [ Wend , Lemma 4.4.12]), one can find a smooth Since the present discussion is purely linear, it does not require the assumption kp > ectures on Symplectic Field Theory section Y ∈ T J J U whose value at u ( z ) is chosen such that Y ◦ T u ◦ j = θ at z , so their pointwise inner product is positive in some neighborhood of z . Butby Theorem B.6, one can multiply a small perturbation of Y by a bump functionto produce a section (still denoted by Y ) of class C ε so that the pointwise innerproduct of Y ◦ T u ◦ j with θ is positive near z but vanishes everywhere else; notethat this requires the assumption u − ( u ( z )) = { z } , so that the value of Y near u ( z ) affects the value of Y ◦ T u ◦ j near z but nowhere else. This violates thesecond condition in (7.6) and thus completes the proof for k = 1. In the generalcase, suppose θ ∈ W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )). Then θ is also of class L p,δ , sosurjectivity in the k = 1 case implies the existence of η ∈ W ,p,δ and Y ∈ T J J ε U with D u η + Y ◦ T u ◦ j = θ . Since Y ◦ T u ◦ j is smooth with compact support,one can then use elliptic regularity to show η ∈ W k,p,δ , and this proves surjectivityfor arbitrary k ∈ N and p ∈ (1 , ∞ ).The implicit function theorem now implies that whenever kp > B k,p,δ is a well-defined Banach manifold, ¯ ∂ − (0) is a smooth Banach submanifold of T ×B k,p,δ × J ε U in a neighborhood of ( j , u , J ). The projection map¯ ∂ − (0) → J ε U : ( j, u, J ) J is also smooth since it is the restriction to a smooth submanifold of the obviouslysmooth projection map T × B k,p,δ × J ε U → J ε U . Since G acts freely and properlyon ¯ ∂ − (0), the quotient ¯ ∂ − /G then inherits a smooth Banach manifold structurefor which the projection is still smooth, and this quotient is identified locally with M ∗ ( J ε U ). Smoothness of transition maps is shown via the same regularity argumentsas in the proof of Theorem 7.1. (cid:3) Exercise . Show that if X , Y and Z are Banach spaces, T : X → Y is aFredholm operator and A : Z → Y is a bounded linear operator, then the linearmap L : X ⊕ Z → Y : ( x, z ) T x + A z has closed range. Hint: it might help to write X = V ⊕ ker T and Y = W ⊕ coker C so that C ∼ = coker T and V T −→ W is an isomorphism. Exercise . Under the same assumptions as in Exercise 7.16, show that if T is surjective, then L has a bounded right inverse. We claim now that the smooth map(7.7) M ∗ ( J ε U ) → J ε U : ( u, J ) J is a nonlinear Fredholm map, i.e. its derivative at every point is a Fredholm oper-ator. Using the local identification of M ∗ ( J ε U ) with ¯ ∂ − (0) /G as in the proof ofLemma 7.15 and lifting the projection to ¯ ∂ − (0), the derivative of ¯ ∂ − (0) → J ε U at( j , u , J ) takes the formker D ¯ ∂ ( j , u , J ) → T J J ε U : ( y, η, Y ) Y. Chris Wendl
The Fredholm property for this projection is a consequence of the Fredholm propertyfor D u via the following general lemma, whose proof is a routine matter of linearalgebra (cf. [ Wend , Lemma 4.4.13]):
Lemma . Under the assumptions of Exercise 7.16, suppose L is surjective.Then the projection Π : ker L → Z : ( x, z ) z has kernel and cokernel isomorphic to the kernel and cokernel respectively of T : X → Y . (cid:3) By the Sard-Smale theorem, the set of regular values of the projection (7.7) is acomeager subset J ε, reg U ⊂ J ε U , and by Lemma 7.18, every ( u , J ) ∈ M ∗ ( J ε U ) with J ∈ J ε, reg U then has the propertythat D ¯ ∂ J ( j , u ) : T j T ⊕ T u B k,p,δ → E k − ,p,δ ( j ,u ) is surjective, which means u represents a Fredholm regular element of M ( J ). C ε to C ∞ The arguments above would constitute a proof of Theorem 7.2 if we were allowedto replace the space of smooth almost complex structures J U with the space J ε U of C ε -small perturbations of J ref . Let us define J reg U ⊂ J U to be the space of all J ∈ J U with the property that all curves in M ( J ) that haveinjective points mapping to U are Fredholm regular. The theorem claims that thisset is comeager in J U . We can already see at this point that it is dense: indeed, theBaire category theorem implies that J ε, reg U is dense in J ε U , so in particular there existsa sequence J ν ∈ J ε, reg U that converges in to J ref in the C ε -topology and therefore alsoin the C ∞ -topology. The choice of J ref ∈ J U in this discussion was arbitrary, so thisproves density.To prove that J reg U is not only dense but also contains a countable intersection of open and dense sets in J U , we can adapt an argument originally due to Taubes. Theidea is to present the sets of somewhere injective curves in M ( J ) as countable unionsof compact subsets M ∗ N ( J ) for N ∈ N , and thus present J reg U as a correspondingcountable intersection of spaces J reg ,N U that achieve regularity only for the elementsin M ∗ N ( J ). The compactness of M ∗ N ( J ) will then permit us to prove that J reg ,N U isnot only dense but also open.The definition of M ∗ N ( J ) is motivated in part by the knowledge that spaces of J -holomorphic curves have natural compactifications. We have not yet discussedthe compactification M ( J ) of M ( J ), but we have covered enough of the analyticaltechniques behind this construction to suffice for the present discussion. Recall firstthat the moduli space of Riemann surfaces M g,ℓ of genus g with ℓ marked points ectures on Symplectic Field Theory also has a natural compactification whenever 2 g + ℓ ≥
3, known as the
Deligne-Mumford compactification M g,ℓ ⊃ M g,ℓ . The space M g,ℓ consists of “nodal” Riemann surfaces, which can be understood asobjects that arise from smooth Riemann surfaces with pair-of-pants decompositionsin the limit where some of the lengths of the circles separating two pairs of pantsfrom each other may degenerate to 0 (see e.g. [ SS92 ]). We will discuss this in abit more detail in Lecture 9; for now, all you really need to know is that M g,ℓ isa compact and metrizable topological space that contains M g,ℓ as an open subset.Let us fix a metric on M g,ℓ and denote the distance function by dist( , ).Similarly, fix Riemannian metrics on c W and ˙Σ with translation-invariance onthe cylindrical ends and use dist( , ) to denote the distance functions. For N ∈ N and J ∈ J U , we define M ∗ N ( J ) ⊂ M ( J )to be the set of equivalence classes admitting representatives (Σ , j, Γ + , Γ − , Θ , u ) withthe following properties: • The equivalence class in M g,k + + k − + m represented by (Σ , j, Γ + ∪ Γ − ∪ Θ) liesat a distance of at most 1 /N from M g,k + + k − + m \ M g,k + + k − + m ; • sup z ∈ ˙Σ | du ( z ) | ≤ N ; • There exists z ∈ ˙Σ such thatdist( u ( z ) , c W \ U ) ≥ N , | du ( z ) | ≥ N , and inf z ∈ ˙Σ \{ z } dist( u ( z ) , u ( z ))dist( z , z ) ≥ N .
We observe that every element of M ∗ N ( J ) has an injective point mapped into U , andconversely, every asymptotically cylindrical J -holomorphic curve with that propertybelongs to M ∗ N ( J ) for N ∈ N sufficiently large. It is crucial to observe that all threeconditions in this definition are closed conditions: morally, we are defining M ∗ N ( J )to be a closed subset in the compactification of M ( J ), and it will therefore becompact.Define J reg ,N U ⊂ J U as the set of all J ∈ J U for which every element of M ∗ N ( J ) is Fredholm regular. Lemma . For every N ∈ N , J reg ,N U is open and dense. Proof.
Density is immediate, since we’ve seen already that every J ∈ J U admitsa C ∞ -small perturbation that achieves regularity for all curves in S N ∈ N M ∗ N ( J ).For openness, suppose the contrary: then there exists J ∞ ∈ J reg ,N U and a sequence If the stability condition 2 g + k + + k − + m ≥ , j, Γ + ∪ Γ − , Θ ′ ), where Θ ′ is the unionof Θ with enough extra marked points to achieve stability. Chris Wendl J ν ∈ J U \J reg ,N U with J ν → J ∞ in the C ∞ -topology. There must also exist a sequenceof curves u ν ∈ M ∗ N ( J ν ) that are not Fredholm regular. By the definition of M ∗ N ( J ν ),they have domains that are uniformly bounded away from the singular part of theDeligne-Mumford space of Riemann surfaces, so we can extract a subsequence forwhich these domains converge. Similarly, the first derivatives of u ν are uniformlybounded, implying in particular a uniform W ,p -bound locally for some p >
2, andelliptic regularity (Theorem 2.22 in Lecture 2) turns this into uniform C ∞ -boundand thus a C ∞ -convergent subsequence u ν → u ∞ ∈ M ∗ N ( J ∞ ). But u ∞ must thenbe Fredholm regular, which is an open condition, implying that u ν is also regularfor ν sufficiently large, and this is a contradiction. (cid:3) Proof of Theorem 7.2.
Since the space of all curves in M ( J ) with injectivepoints mapped into U is the union of the spaces M ∗ N ( J ) for N ∈ N , we have J reg U = \ N ∈ N J reg ,N U , which is a countable intersection of open and dense sets. (cid:3) ECTURE 8
Transversality in symplectizationsContents
This lecture is an addendum to the transversality discussion in Lecture 7: weneed to prove that Fredholm regularity can also be achieved for generic translation-invariant almost complex structures on symplectizations.
Theorem 7.2 in the previous lecture stated that generic perturbations of J in aprecompact open subset U of a completed symplectic cobordism suffice to achieveregularity for all simple holomorphic curves that pass through that subset. In themore specialized setting of a symplectization R × M with an R -invariant almostcomplex structure J ∈ J ( H ), we need a more specialized transversality result, asthe generic perturbation from Theorem 7.2 cannot be expected to stay in the space J ( H ), in particular it will usually not be R -invariant. The following statement refersto a stable Hamiltonian structure H = ( ω, λ ) with induced hyperplane distribution ξ = ker λ and Reeb vector field R , and we denote by π ξ : T ( R × M ) → ξ the projection along the trivial subbundle generated by ∂ r and R . We assume asusual that M ( J ) denotes a moduli space of asymptotically cylindrical J -holomorphiccurves with a fixed genus and number of marked points, representing a fixed relativehomology class and asymptotic to fixed sets of nondegenerate Reeb orbits at itspositive and negative punctures. Theorem . Suppose M is a closed (2 n − -dimensional manifold carrying astable Hamiltonian structure H = ( ω, λ ) , J fix ∈ J ( H ) , and U ⊂ M is an open subset. Then there exists a comeager subset J reg U ⊂ (cid:8) J ∈ J ( H ) (cid:12)(cid:12) J = J fix on R × ( M \ U ) (cid:9) such that for every J ∈ J reg U , every curve u ∈ M ( J ) with a representative u : ˙Σ → R × M that has an injective point z ∈ ˙Σ satisfying Chris Wendl (i) u ( z ) ∈ R × U ,(ii) π ξ ◦ du ( z ) = 0 , and(iii) im ( π ξ ◦ du ( z )) ∩ ker ( dλ | ξ ) = { } is Fredholm regular. This result is applied most frequently with U = M , in which case the condition u ( z ) ∈ R × U is vacuous. The second and third conditions on the injective point z can be rephrased by asking for the linear map dλ ( π ξ T u ( X ) , · ) | ξ u ( z ) : ξ u ( z ) → R to be nontrivial for every nonzero X ∈ T z ˙Σ. If λ is contact, then this is immediatewhenever π ξ T u ( X ) = 0 since dλ | ξ is nondegenerate, and the condition π ξ T u ( X ) = 0is also easy to achieve: Proposition . If J ∈ J ( H ) , then for any connected J -holomorphic curve u : ( ˙Σ , j ) → ( R × M, J ) , the section π ξ ◦ du ∈ Γ(Hom C ( T ˙Σ , u ∗ ξ )) either is identically zero or has only isolated zeroes. As you might guess, this result is a consequence of the similarity principle; see § π ξ ◦ du ≡
0, then u is everywhere tangent to thevector fields ∂ r and R , so if it is asymptotically cylindrical, then it can only be atrivial cylinder or a cover thereof. Proposition . All trivial cylinders over nondegenerate Reeb orbits have in-dex and are Fredholm regular. Proof.
Let u γ : R × S → R × M denote the trivial cylinder over an orbit γ : S → M . The virtual dimension formula proved in Lecture 7 givesind( u γ ) = ( n − χ ( R × S ) + 2 c τ ( u ∗ γ T ( R × M )) + µ τ CZ ( γ ) − µ τ CZ ( γ )= 2 c τ ( u ∗ γ T ( R × M )) = 0since the asymptotic trivialization τ has an obvious extension to a global trivial-ization of u ∗ γ ξ , and u ∗ γ T ( R × M ) is globally the direct sum of the latter with thetrivial line bundle spanned by ∂ r and R . Using this splitting, the linearized Cauchy-Riemann operator D u γ can be identified with ¯ ∂ ⊕ ( ∂ s − A γ ), where¯ ∂ = ∂ s + i∂ t : W k,p,δ ( R × S , C ) ⊕ V Γ → W k − ,p,δ ( R × S , C )and ∂ s − A γ : W k,p,δ ( u ∗ γ ξ ) → W k − ,p,δ ( u ∗ γ ξ ) . Here we are assuming without loss of generality that V Γ is a complex 2-dimensionalspace of smooth sections of the trivial line bundle spanned by ∂ r and R that areconstant near infinity, and we are identifying this with a space of smooth complex-valued functions on R × S . Nondegeneracy implies that ∂ s − A : W k,p → W k − ,p is an isomorphism, recall Theorem 4.11 in Lecture 4. Using weight functions asin the proof of Lemma 7.10 to define isomorphisms between W k,p,δ and W k,p , onecan identify ∂ s − A γ : W k,p,δ → W k − ,p,δ with a small perturbation of the same ectures on Symplectic Field Theory operator W k,p → W k − ,p , hence it is also an isomorphism for δ > ∂ : W k,p,δ ⊕ V Γ → W k − ,p,δ is also surjective, observe first thatits index is 2; this follows from our calculation of ind( u γ ) and corresponds to thefact that dim Aut( R × S , i ) = 2. The kernel of this operator consists of boundedholomorphic C -valued functions on R × S , so it is precisely the real 2-dimensionalspace of constant functions, implyingdim R coker( ¯ ∂ ) = dim R ker( ¯ ∂ ) − ind R ( ¯ ∂ ) = 2 − , so D u γ is surjective. (cid:3) Corollary . For any contact form α on a closed manifold M , there existsa comeager subset J reg ( α ) ⊂ J ( α ) such that for every J ∈ J reg ( α ) , all somewhereinjective asymptotically cylindrical J -holomorphic curves in R × M are Fredholmregular. (cid:3) Note that in the setting of Corollary 8.4, a curve that is not a cover of a trivialcylinder always belongs to a smooth 1-parameter family of curves related to eachother by R -translation, so that the kernel of the linearized Cauchy-Riemann operatorautomatically has kernel of dimension at least 1. This precludes Fredholm regularityfor curves of index 0, thus: Corollary . If α is a contact form and J ∈ J reg ( α ) , then all simple asymp-totically cylindrical J -holomorphic curves u : ( ˙Σ , j ) → ( R × M, J ) other than trivialcylinders satisfy ind( u ) ≥ . (cid:3) The following example shows that the third condition on the injective point inTheorem 8.1 cannot be fully removed in general.
Example . Assume (
W, ω ) is a closedsymplectic manifold of dimension 2 n − H : S × W → R , and M := S × W is assigned the stable Hamiltonian structure(Ω , Λ) := ( ω + dt ∧ dH, dt ). A choice of J ∈ J ( H ) is then equivalent to a choice of t -dependent family of ω -compatible almost complex structures { J t ∈ J ( W, ω ) } t ∈ S ,and for any t ∈ S and s ∈ R , J t -holomorphic curves u : (Σ , j ) → ( W, J t ) give riseto J -holomorphic curves¯ u : (Σ , j ) → ( R × M, J ) : z ( s, t, u ( z )) . In particular, when n = 2 one can consider the example where W = Σ is a closedsurface, so curves of this form exist for any choice of J ∈ J ( H ), no matter howgeneric (remember that the domain complex structure j is arbitrary, it is not fixedin advance). If Σ has genus g and the map u : Σ → Σ has degree 1, then since ¯ u has no punctures and satisfies c ([¯ u ]) = c (¯ u ∗ T ( R × S × Σ)) = c ( T Σ) = χ (Σ), theindex of ¯ u is ind(¯ u ) = ( n − χ (Σ) + 2 χ (Σ) = χ (Σ) = 2 − g. This shows that ¯ u cannot be Fredholm regular unless g = 0. Chris Wendl
Theorem 8.1 appeared for the first time in the contact case in [
Dra04 ], andalternative proofs have since appeared in the appendix of [
Bou06 ] (for cylinders inthe contact case) and in [
Wena ] (under slightly different assumptions in the stableHamiltonian setting). What I will describe below is a generalization of Bourgeois’sproof.
One point of difficulty in proving transversality in R × M is that in contrast tothe setting of Theorem 7.2, generic perturbations within J ( H ) can never be trulylocal, i.e. if you perturb J near a point ( r, x ) ∈ R × M , then you are also perturbing itin a neighborhood of the entire line R × { x } . We therefore need to know that we canfind a point z ∈ ˙Σ that is the only point where u : ˙Σ → R × M passes through sucha line; put another way, we need to know that not only u = ( u R , u M ) : ˙Σ → R × M but also the projected map u M : ˙Σ → M is somewhere injective. The first step inshowing this is Proposition 8.2 above, as the zeroes of the section π ξ ◦ du ∈ Γ(Hom C ( T ˙Σ , u ∗ ξ ))are precisely the critical points of u M : ˙Σ → M ; everywhere else, u M is an immersiontransverse to the Reeb vector field. To prove Proposition 8.2, we shall use the factthat the vector fields ∂ r and R generate an integrable J -invariant distribution on R × M . Indeed, the zeroes of π ξ ◦ du are the points of tangency with this distribution,hence the result is an immediate consequence of the following statement: Lemma . Suppose ( W, J ) is an almost complex manifold, Ξ ⊂ T W is a smoothintegrable J -invariant distribution and u : (Σ , j ) → ( W, J ) is a connected pseudo-holomorphic curve whose image is not contained in a leaf of the foliation generatedby Ξ . Then all points z ∈ Σ with im du ( z ) ⊂ Ξ are isolated in Σ . Proof.
Statement is local, so assume (Σ , j ) = ( D , i ) with coordinates s + it , W = C n , and u (0) = 0. Let 2 m denote the real dimension of Ξ, and observe thatsince Ξ is integrable, we can change coordinates near 0 and assume without loss ofgenerality that at every point p ∈ C n near 0, Ξ p = C m ⊕ { } ⊂ C n = T p C n . The J -invariance of Ξ then implies that in coordinates ( w, ζ ) ∈ C m × C n − m , J takes theform J ( w, ζ ) = (cid:18) J ( w, ζ ) Y ( w, ζ )0 J ( w, ζ ) (cid:19) , where J and J are both − , and J Y + Y J = 0. Writing u ( z ) = ( f ( z ) , v ( z )) ∈ C m × C n − m , the Cauchy-Riemann equation ∂ s u + J ( u ) ∂ t u = 0 is then equivalent tothe two equations ∂ s f + J ( f, v ) ∂ t f + Y ( f, v ) ∂ t v = 0 ,∂ s v + J ( f, v ) ∂ t v = 0 . (8.1)We have im du ( z ) ⊂ Ξ wherever ∂ s v = ∂ t v = 0; notice that it suffices to considerthe condition ∂ s v = 0 since ∂ t v = J ( f, v ) ∂ s v . Differentiating the second equationin (8.1) with respect to s gives ∂ s ( ∂ s v ) + J ( f, v ) ∂ t ( ∂ s v ) + ∂ s [ J ( f, v )] J ( f, v ) ∂ s v = 0 , ectures on Symplectic Field Theory where in the last term we’ve substituted J ( f, v ) ∂ s v for ∂ t v . Setting ¯ J ( z ) := J ( f ( z ) , v ( z )) and A ( z ) := ∂ s [ J ( f ( z ) , f ( z )] J ( f ( z ) , v ( z )), this becomes a linearCauchy-Riemann type equation ∂ s ( ∂ s v ) + ¯ J ∂ t ( ∂ s v ) + A ( ∂ s v ) = 0, so the similarityprinciple implies that zeroes of ∂ s v are isolated unless it is identically zero. Thelatter would mean v is constant, so u is contained in a leaf of Ξ. (cid:3) Lemma . Suppose J ∈ J ( H ) , γ : S → M is a closed Reeb orbit, and u = ( u R , u M ) : ( ˙Σ , j ) → ( R × M, J ) is an asymptotically cylindrical J -holomorphiccurve that is not a cover of a trivial cylinder. Then all intersections of the map u M : ˙Σ → M with the image of the orbit γ are isolated. Proof.
The trivial cylinder over γ is a J -holomorphic curve, so the statementfollows from the fact that two asymptotically cylindrical J -holomorphic curves canonly have isolated intersections unless both are covers of the same simple curve. (cid:3) We can now prove the statement we need about somewhere injectivity for u M :˙Σ → M . This result first appeared in [ HWZ99 , Theorem 1.13].
Proposition . Suppose J ∈ J ( H ) and u = ( u R , u M ) : ( ˙Σ , j ) → ( R × M, J ) is a simple asymptotically cylindrical J -holomorphic curve which is not a trivialcylinder and has only nondegenerate asymptotic orbits. Then the set of injectivepoints z ∈ ˙Σ of the map u M : ˙Σ → M for which u M ( z ) is not contained in any ofthe asymptotic orbits of u is open and dense. Proof.
Openness is clear, so our main task is to prove density. The idea isfirst to show via elementary topological arguments that if the set of injective pointsis not dense, then ˙Σ contains two disjoint open sets on which u M is an embeddingwith identical images. We will then conclude from this that if u is simple, it mustbe equivalent to one of its nontrivial R -translations, and the latter is impossible foran asymptotically cylindrical curve. Step 1 : We begin by harmlessly removing some discrete sets of points in ˙Σ thatwould make the subsequent arguments more complicated. Let P ⊂ M denote the union of the images of the asymptotic orbits of u , a finite disjoint union ofcircles. Lemma 8.8 implies that u − M ( P ) is a discrete subset of ˙Σ. By Proposition 8.2,there is also a discrete set Z ⊂ ˙Σ \ u − M ( P ) containing all points z u − M ( P ) where π ξ ◦ du ( z ) = 0, and we claim that Z ′ := u − M ( u M ( Z ))is a discrete subset of ˙Σ \ u − M ( P ). Indeed, u M ( Z ) is a discrete subset of M \ P since the points in Z can only accumulate at infinity, hence accumulation points of u M ( Z ) ⊂ M can occur only in P . For each individual point p ∈ u M ( Z ), the fact that p P implies u − M ( p ) is compact, and it consists of a discrete (and therefore finite) Actually the asymptotic formula of [
HWZ96 ] implies that both Z and u − M ( P ) are alwaysfinite for curves that are not covers of trivial cylinders, but we do not need to use that here. Chris Wendl set of points with π ξ ◦ du ( z ) = 0, plus possibly some other points where π ξ ◦ du ( z ) = 0,but u M is an embedding near each point of the latter type, so that these points of u − M ( p ) must always be isolated and are therefore also finite in number. This provesthe claim, and we conclude that¨Σ := ˙Σ \ (cid:0) u − M ( P ) ∪ Z ′ (cid:1) an open and dense subset of ˙Σ, as it is obtained by removing a discrete subset fromthe open and dense subset ˙Σ \ u − M ( P ). To prove the proposition, it will now sufficeto prove that the set of points z ∈ ¨Σ which are injective points of u M : ˙Σ → M isdense in ¨Σ. We shall argue by contradiction and assume from now on that densityfails. Step 2 : We will find two open subsets U , V ⊂ ˙Σ such that u M restricts to anembedding on both, but U ∩ V = ∅ and u M ( U ) = u M ( V ) . Indeed, assume the set of injective points of u M lying in ¨Σ is not dense in ¨Σ. Thenthere exists a point z ∈ ¨Σ with a closed neighborhood D ( z ) ⊂ ¨Σ such that no z ∈ D ( z ) is an injective point. Since z ∈ ¨Σ implies π ξ ◦ du ( z ) = 0, this means thatfor every z ∈ D ( z ), there exists ζ ∈ ˙Σ \ { z } with u M ( z ) = u M ( ζ ), and the definitionof ¨Σ implies ζ is also in ¨Σ, hence π ξ ◦ du ( ζ ) = 0 and u M is a local embedding near ζ .Since u ( z ) P and u M maps ˙Σ \ u − M ( P ) properly to M \ P , we also conclude that u − M ( u M ( z )) is finite. Now suppose u − M ( u M ( z )) = { z , ζ , . . . , ζ m } , and let D ( ζ j ) ⊂ ¨Σfor j = 1 , . . . , m denote closed neighborhoods on which u M is an embedding. Weclaim that after possibly shrinking D ( z ), we can assume u M ( D ( z )) ⊂ m [ j =1 u M ( D ( ζ j ) . Let us first shrink D ( z ) so that u M is an embedding on D ( z ), which is possiblesince π ξ ◦ du ( z ) = 0. Then if the claim is false, there exists a sequence z ν ∈ D ( z ) ofnoninjective points with z ν → z , hence there is also a sequence z ′ ν ∈ ¨Σ \ D ( z ) with u M ( z ν ) = u M ( z ′ ν ) but z ′ ν not converging to any of ζ , . . . , ζ m . But since u M ( z ′ ν ) → u M ( z ) P , the points z ′ ν are confined to a compact subset of ˙Σ and therefore havea subsequence z ′ ν → z ′∞ ∈ ˙Σ with u M ( z ′∞ ) = u M ( z ). The limit cannot be z itselfsince z ′ ν
6∈ D ( z ), thus z ′∞ must be one of the ζ , . . . , ζ m , and we have a contradiction.We claim next that at least one of the sets u M ( D ( z )) ∩ u M ( D ( ζ j )) has nonemptyinterior. This is a simple exercise in metric space topology: it can be reduced tothe fact that if X is a metric space with closed subsets V, W ⊂ X that both haveempty interior (meaning no open subset of X is contained in V or W ), then V ∪ W also has empty interior. Since the subsets u M ( D ( z )) ∩ u M ( D ( ζ j )) ⊂ u M ( D ( z )) for j = 1 , . . . , m are all closed but their union is u M ( D ( z )), they cannot all have emptyinterior. This achieves the goal of Step 2. Step 3 : We show that u is biholomorphically equivalent to one of its R -translations τ · u := ( u R + τ, u M ) : ˙Σ → R × M ectures on Symplectic Field Theory for τ ∈ R \ { } . To see this, note that for J ∈ J ( H ), the nonlinear Cauchy-Riemannequation T u ◦ j = J ( u ) ◦ T u is equivalent to the two equations du R = u ∗ M λ ◦ j,π ξ ◦ T u M ◦ j = J ( u M ) ◦ π ξ ◦ T u M . (8.2)Since π ξ ◦ T u M : ˙Σ → u ∗ M ξ is injective everywhere on the neighborhoods U and V , thesecond equation determines j in terms of J on each of these regions; in particular,the identification of u M ( U ) with u M ( V ) provides a biholomorphic map of V to U so that u | U and u | V may be regarded as two J -holomorphic maps from the sameRiemann surface which differ only in the R -factor. But with j and u M both fixed,the first equation in (8.2) determines du R and thus determines u R up to the additionof a constant τ ∈ R . If τ = 0, this means u has two disjoint regions on which itsimages are identical, contradicting the assumption that u is simple. Thus τ = 0,and since two distinct simple curves can only intersect each other at isolated points,we conclude u = τ · u up to parametrization. Step 4 : We now derive a contradiction. The relation u = τ · u implies that infact u = kτ · u for every k ∈ Z , so we obtain a diverging sequence of R -translations τ k → ∞ such that u and τ k · u always have identical images in R × M . It followsthat for some point z ∈ ˙Σ with u ( z ) = ( r, x ) where x is not contained in any of theasymptotic orbits of u , the points ( r − τ k , x ) are all in the image of u as τ k → ∞ .But this contradicts the asymptotically cylindrical behavior of u . (cid:3) The overall outline of the proof of Theorem 8.1 is the same as for Theorem 7.2:one needs to define a suitable space J ε U of perturbed almost complex structures,giving rise to a universal moduli space M ∗ ( J ε U ) that is a smooth Banach manifold,and then apply the Sard-Smale theorem to conclude that generic elements of J ε U are regular values of the projection M ∗ ( J ε U ) → J ε U : ( u, J ) J . If J ε U is a spaceof C ε -perturbed almost complex structures, then in the final step one can use theTaubes trick as in § J ε U into a genericityresult within the space J ( H ) of smooth almost complex structures. The only stepthat differs meaningfully from what we’ve already discussed is the smoothness ofthe universal moduli space, so let us focus on this detail.Assume J ref ∈ J ( H ) with J ref = J fix outside R ×U , and J ε U is a Banach manifoldof C ε -small perturbations of J ref in J ( H ) that are also fixed outside of R × U . Therelevant universal moduli space is then defined by M ∗ ( J ε U ) := (cid:8) ( u, J ) (cid:12)(cid:12) J ∈ J ε U , u ∈ M ( J ) and u : ˙Σ → R × M has an injective point z ∈ ˙Σ with u ( z ) ∈ R × U and im ( π ξ ◦ du ( z )) ∩ ker ( dλ | ξ ) = { } (cid:9) . Notice that both of the constraints satisfied by u at the injective point are open.The local structure of M ∗ ( J ε U ) near an element ( u , J ) with representative u :( ˙Σ , j ) → ( R × M, J ) can again be described via the zero set of a smooth section¯ ∂ : T × B k,p,δ × J ε U → E k − ,p,δ : ( j, u, J ) T u ◦ J ◦ T u ◦ j, Chris Wendl where T is a Teichm¨uller slice through j , and it suffices to show that the lineariza-tion L : T u B k,p,δ ⊕ T J J ε U → E k − ,p,δ ( j ,u ,J ) : ( η, Y ) D u η + Y ◦ T u ◦ j is always surjective. As usual, here we’re assuming k ∈ N , 1 < p < ∞ , and theexponential weight δ > D u is Fredholm. The imageof L is then closed, and focusing on the k = 1 case, if L is not surjective then thereexists a nontrivial element θ ∈ L q, − δ (Hom C ( T ˙Σ , u ∗ T ( R × M ))) such that h D u η, θ i L = 0 for all η ∈ W ,p,δ ( u ∗ T ( R × M )) ⊕ V Γ , h Y ◦ T u ◦ j , θ i L = 0 for all Y ∈ T J J ε U . (8.3)The first condition implies via elliptic regularity and the similarity principle that θ is smooth and has only isolated zeroes. So far this is all the same as in the proofof Theorem 7.2, but the next step is trickier: since perturbing J within J ( H )only changes the action of the almost complex structure on ξ but not on the trivialsubbundle generated by ∂ r and R , it is not clear whether the range of values allowedfor Y is large enough to force h Y ◦ T u ◦ j , θ i L > T ( R × M ) = ǫ ⊕ ξ, where ǫ denotes the trivial line bundle spanned by ∂ r and R . In particular, thedomain and target bundles of the Cauchy-Riemann type operator D u now split as u ∗ T ( R × M ) = u ∗ ǫ ⊕ u ∗ ξ, Hom C ( T ˙Σ , u ∗ T ( R × M )) = Hom C ( T ˙Σ , u ∗ ǫ ) ⊕ Hom C ( T ˙Σ , u ∗ ξ ) , and we shall write η = ( η ǫ , η ξ ) and θ = ( θ ǫ , θ ξ ) accordingly. This gives a blockdecomposition of D u as D u η = (cid:18) ( D u η ) ǫ ( D u η ) ξ (cid:19) = (cid:18) D ǫu D ǫξu D ξǫu D ξu (cid:19) (cid:18) η ǫ η ξ (cid:19) . It is easy to verify that D ǫu and D ξu each satisfy suitable Leibniz rules and are thusCauchy-Riemann type operators on u ∗ ǫ and u ∗ ξ respectively, while the off-diagonalterms are both tensorial, i.e. zeroth-order operators. Since perturbations of J in J ( H ) only change its action on ξ , Y ∈ T J J ε U now takes the block form Y = (cid:18) Y ξ (cid:19) , where Y ξ is a C ε -small section of the bundle End C ( ξ, J ) over M . Assuming the L -pairings are defined so as to respect these splittings, the second condition in (8.3)now becomes h Y ξ ◦ π ξ ◦ T u ◦ j , θ ξ i L = 0 , and given any injective point z ∈ ˙Σ of ( u ) M : ˙Σ → M satisfying u ( z ) ∈ R × U , wehave enough freedom to choose Y ξ near R × { u ( z ) } such that this pairing becomespositive unless θ ξ = 0 near z . ectures on Symplectic Field Theory It remains to show that θ ǫ also vanishes near z , which will contradict the fact that θ only has isolated zeroes. To this end, notice that the first condition in (8.3) impliesvia separate choices of the components η ǫ and η ξ with support near z that h D ǫu η ǫ , θ ǫ i L = 0 for all η ǫ supported near z , h D ǫξu η ξ , θ ǫ i L = 0 for all η ξ supported near z . (8.4)The first of these two conditions gives no new information, since we already knowthat θ = ( θ ǫ ,
0) solves an anti-Cauchy-Riemann equation. To get some informationout of the second condition, we will need an explicit formula for D ǫξu . Lemma . The tensorial operator D ǫξu : u ∗ ξ → Hom C ( T ˙Σ , u ∗ ǫ ) takes the form D ǫξu η ξ = h − dλ (cid:0) η ξ , J ξ ◦ π ξ ◦ T u ( · ) (cid:1)i ∂ r + (cid:2) dλ (cid:0) η ξ , π ξ ◦ T u ( · ) (cid:1)(cid:3) R. Proof.
As a preliminary step, notice that − dr ◦ J = λ for any J ∈ J ( H );indeed, the conditions J ( ξ ) = ξ ⊂ ker dr and J ∂ r = R imply that these two 1-formshave matching values on ∂ r , R and ξ . As a consequence, λ ◦ J = dr , so in particular λ ◦ J is closed.Choosing local holomorphic coordinates ( s, t ) in an arbitrary neighborhood in˙Σ, we have ( D ǫξu η ξ ) ∂ s = dr (cid:0) ( D u η ξ ) ∂ s (cid:1) ∂ r + λ (cid:0) ( D u η ξ ) ∂ s (cid:1) R. Extend u : ˙Σ → R × M to a smooth 1-parameter family of maps { u ρ : ˙Σ → R × M } ρ ∈ R with ∂ ρ u ρ | ρ =0 = η ξ ∈ Γ( u ∗ ξ ). Then by the definition of the linearizedCauchy-Riemann operator,( D u η ξ ) ∂ s = ∇ ρ ( ∂ s u ρ + J ( u ρ ) ∂ t u ρ ) | ρ =0 , for any choice of connection ∇ on R × M . Since ∂ s u + J ( u ) ∂ t u = 0, we find λ (cid:0) ( D u η ξ ) ∂ s (cid:1) = λ (cid:0) ∇ ρ ( ∂ s u ρ + J ( u ρ ) ∂ t u ρ ) | ρ =0 (cid:1) = ∂ ρ [ λ ( ∂ s u ρ + J ( u ρ ) ∂ t u ρ )] | ρ =0 = ∂ ρ [ λ ( ∂ s u ρ )] | ρ =0 + ∂ ρ [( λ ◦ J )( ∂ t u ρ )] | ρ =0 = dλ ( η ξ , ∂ s u ) + d ( λ ◦ J )( η ξ , ∂ t u )= dλ ( η ξ , π ξ ∂ s u ) , where we’ve used the formula dλ ( X, Y ) = L X [ λ ( Y )] − L Y [ λ ( X )] − λ ([ X, Y ])and eliminated several terms using the fact that λ ( η ξ ) = λ ( J η ξ ) = 0 since η ξ isvalued in ξ , plus d ( λ ◦ J ) = 0. A similar computation gives dr (cid:0) ( D u η ξ ) ∂ s (cid:1) = − dλ ( η ξ , π ξ ∂ t u ) = − dλ ( η ξ , J ◦ π ξ ∂ s u ) , so removing the local coordinates from the picture produces the stated formula. (cid:3) The following exercise in symplectic linear algebra shows that this bundle map u ∗ ξ → Hom C ( T ˙Σ , u ∗ ǫ ) is surjective on any fiber over a point z with π ξ ◦ du ( z ) =0. (If you have no patience for the exercise, just convince yourself that it’s truewhenever dλ | ξ is nondegenerate and tames J | ξ , i.e. the contact case.) Chris Wendl
Exercise . Assume V is a finite-dimensional vector space, X, Y ⊂ V arelinearly independent vectors, and Ω is an alternating bilinear form on V . Show thatthe real-linear map A : V → C : v Ω( v, X ) + i Ω( v, Y )is surjective if and only if Span( X, Y ) ∩ ker Ω = { } . Hint: Under the latter condition, one loses no generality by replacing V with asubspace that is complementary to ker Ω and contains Span(
X, Y ) , in which case ( V, Ω) becomes a symplectic vector space. Now consider the restriction of A to a -dimensional subspace transverse to the symplectic complement of Span(
X, Y ) . The conclusion of this discussion is that unless θ ǫ vanishes near z , η ξ can bechosen with support near z so that h D u η ξ , θ ǫ i L >
0, violating the second condi-tion in (8.4). This proves that θ vanishes altogether near z and thus, by uniquecontinuation, θ ≡
0, a contradiction.We’ve proved that the universal moduli space is smooth as claimed. Since therest of the proof of Theorem 8.1 is the same as in the non- R -invariant case, we leavethose details to the reader. Remark . You may have noticed that in both this and the previous lecture,our proof that the universal moduli space is smooth relied on a surjectivity resultthat was actually stronger than needed: in both cases, we needed to prove that anoperator of the form T j T ⊕ T u B k,p,δ ⊕ T J J ε U L −→ E k − ,p,δ ( j ,u ,J ) was surjective, but we ended up proving that its restriction to the smaller domain T u B k,p,δ ⊕ T J J ε U is already surjective. This technical detail hints at a stronger resultthat can be proved using these methods: one can show that not only is M ∗ ( J ε U )smooth but also the forgetful map M ∗ ( J ε U ) → M g,k + + k − + m ([(Σ , j, Γ + , Γ − , Θ , u )] , J ) [(Σ , j, Γ + ∪ Γ − ∪ Θ)]sending a J -holomorphic curve to its underlying domain in the moduli space ofRiemann surfaces is a submersion, cf. the blog post [ Wenb ] and its sequel. One canuse this to prove generic transversality results for spaces of J -holomorphic curveswhose domains are constrained within the moduli space of Riemann surfaces, whichcan be used to define more elaborate algebraic structures on SFT, e.g. this ideaplays a very prominent role in the study of Gromov-Witten invariants.ECTURE 9 Asymptotics and compactnessContents
Moduli spaces of pseudoholomorphic curves are generally not compact, but theyhave natural compactifications , obtained by allowing certain types of curves withsingular behavior. For closed holomorphic curves, this fact is known as
Gromov’scompactness theorem , and our main goal in this lecture is to state its generalizationto punctured curves, which is usually called the
SFT compactness theorem . The the-orem was first proved in [
BEH + ] (see also [ CM05 ] for an alternative approach),and we do not have space here to present a complete proof, but we can still describethe main geometric and analytical ideas behind it.The overarching theme of this lecture is the notion of bubbling , of which we willsee several examples. Bubbling arises in a natural way from elliptic regularity: recallthat in Lecture 2, we proved that whenever kp >
2, any uniformly W k,p -boundedsequence u ν of holomorphic curves is also uniformly C m loc -bounded for every m ≥ N (cf. Theorem 2.22). The Arzel`a-Ascoli theorem implies that such sequences have C ∞ loc -convergent subsequences, and this is true in particular whenever u ν is uniformly C -bounded, as a C -bound implies a W ,p -bound with p >
2. Let us take note ofthis fact for future use:
Proposition . If ( W, J ν ) is a sequence of almost complex manifolds with J ν → J in C ∞ , then any uniformly C -bounded sequence of J ν -holomorphic maps u ν : D → W has a subsequence convergent in C ∞ loc on ˚ D . If one wants to prove compactness for a moduli space of J -holomorphic curves,it therefore suffices in general to establish a C -bound. The catch is, of course,that the first derivatives of u ν might not be uniformly bounded, and this is when Chris Wendl interesting things are seen to happen: while the sequence u ν is not compact, itturns out that it becomes compact after removing finitely many points from itsdomain, and near those points one can take a sequence of reparametrizations to findadditional nontrivial holomorphic curves in the limit, the so-called “bubbles”. Thisis one of the ways that the “nodal” curves in Gromov’s compactness theorem canarise, and we will see the same phenomenon at work in several other contexts aswell. As an important tool for use in the rest of this lecture, we begin with the followingresult from [
Gro85 ]: Theorem . Assume ( W, ω ) is asymplectic manifold with a tame almost complex structure J , and u : D \ { } → W is a J -holomorphic curve that has its image contained in a compact subset of W andsatisfies Z D \{ } u ∗ ω < ∞ . Then u admits a smooth extension to D . We will prove the slightly weaker statement that u has a continuous extension.If dim R W = 2, then the smooth extension follows from this by classical complexanalysis; in higher dimensions, one can instead apply results on local elliptic regu-larity, see e.g. [ MS04 ]. We will use as a black box the following additional resultfrom [
Gro85 ], which is closely related to a standard result about minimal surfaces:
Theorem (Gromov’s monotonicity lemma [
Gro85 ]) . Suppose ( W, ω ) is a com-pact symplectic manifold (possibly with boundary), J is an ω -tame almost complexstructure, and B r ( p ) ⊂ W denotes the open ball of radius r > about p ∈ W withrespect to the Riemannian metric g ( X, Y ) := ω ( X, J Y ) + ω ( Y, J X ) . Then thereexist constants c, R > such that for all r ∈ (0 , R ) and p ∈ W with B r ( p ) ⊂ W , ev-ery proper non-constant J -holomorphic curve u : (Σ , j ) → ( B r ( p ) , J ) passing through p satisfies Z Σ u ∗ ω ≥ cr . In the above statement, (Σ , j ) is assumed to be an arbitrary (generally noncom-pact) Riemann surface without boundary . In applications, one typically has a larger(e.g. closed or punctured) domain Σ ′ in the picture, and Σ is defined to be the con-nected component of u − ( B r ( p )) ⊂ Σ ′ containing some point z ∈ u − ( p ). The mainmessage of the theorem is that u must use up at least a certain amount of energyfor every ball whose center it passes through, so e.g. the portion of the curve passingthrough B r ( p ) cannot become arbitrarily “thin” as in Figure 9.1.Returning to the removable singularity theorem, we shall use the biholomorphicmap Z + := [0 , ∞ ) × S → D \ { } : ( s, t ) e − π ( s + it ) ectures on Symplectic Field Theory (( − B r ( p ) pu (Σ) u (Σ ′ ) Figure 9.1.
The intersection of a J -holomorphic curve u with anopen ball B r ( p ) defines a proper map Σ → B r ( p ). The monotonicitylemma prevents this map from having arbitrarily small area if it passesthrough p .to transform J -holomorphic maps D \ { } → W into maps Z + → W , and the goalwill be to show that whenever such a map u has precompact image and satisfies R Z + u ∗ ω < ∞ , there exists a point p ∈ W such that(9.1) u ( s, · ) → p in C ∞ ( S , W ) as s → ∞ . Fix the obvious flat metric on Z + and any Riemannian metric on W in order todefine norms such as | du ( s, t ) | for ( s, t ) ∈ Z + . Lemma . There exists a constant
C > such that | du ( s, t ) | ≤ C for all ( s, t ) ∈ Z + . Proof, part 1.
Arguing by contradiction, suppose there exists a sequence z k =( s k , t k ) ∈ Z + with | du ( z k ) | =: R k → ∞ . Choose a sequence of positive numbers ǫ k > ǫ k R k → ∞ . We then considerthe sequence of reparametrized maps v k : D ǫ k R k → W : z u ( z k + z/R k ) . These are also J -holomorphic since z z k + z/R k is holomorphic, and the valuesof v k depend only on the values of u over the ǫ k -disk about z k . Notice that since s k → ∞ and ǫ k →
0, we are free to assume that all of these ǫ k -disks are disjoint;moreover, tameness of J implies u ∗ ω ≥ v ∗ k ω ≥
0, thus X k Z D ǫkRk v ∗ k ω = X k Z D ǫk ( z k ) u ∗ ω ≤ Z Z + u ∗ ω < ∞ , implying(9.2) Z D ǫkRk v ∗ k ω → k → ∞ . We would now like to say something about a limit of the maps v k as k → ∞ , but thiswill require a brief pause in the proof, as we don’t yet have quite enough information Chris Wendl to do so. We know that the v k are uniformly C -bounded since u ( Z + ) is containedin a compact subset. It would be ideal if we also had a uniform C -bound, as thenelliptic regularity (Prop. 9.1) would give a C ∞ loc convergent subsequence on the unionof all the domains D ǫ k R k , i.e. on the entire plane. We have dv k ( z ) = 1 R k du ( z k + z/R k ) , hence | dv k (0) | = 1, but we will need to know more about | du | on the rest of D ǫ k ( z k )in order to deduce a C -bound for v k on all of D ǫ k R k . We’ll come back to this in amoment. proof to be continued. . . Here is the auxiliary lemma that is needed to complete the above proof:
Lemma . Suppose ( X, d ) is a complete metric space, g : X → [0 , ∞ ) is continuous, x ∈ X and ǫ > . Then there exist x ∈ X and ǫ > such that,(a) ǫ ≤ ǫ ,(b) g ( x ) ǫ ≥ g ( x ) ǫ ,(c) d ( x, x ) ≤ ǫ , and(d) g ( y ) ≤ g ( x ) for all y ∈ B ǫ ( x ) . Proof.
If there is no x ∈ B ǫ ( x ) such that g ( x ) > g ( x ), then we canset x = x and ǫ = ǫ and are done. If such a point x does exist, then we set ǫ := ǫ / x , ǫ ): that is, if there isno x ∈ B ǫ ( x ) with g ( x ) > g ( x ), we set ( x, ǫ ) = ( x , ǫ ) and are finished, andotherwise define ǫ = ǫ / x , ǫ ). This process must eventuallyterminate, as otherwise we obtain a Cauchy sequence x n with g ( x n ) → ∞ , which isimpossible if X is complete. (cid:3) Proof of Lemma 9.3, part 2.
Applying Lemma 9.4 to X = Z + with g ( z ) = | du ( z ) | , we can replace the original sequences ǫ k and z k with new sequences for whichall the previously stated properties still hold, but additionally, | du ( z ) | ≤ | du ( z k ) | for all z ∈ D ǫ k ( z k ) . Our sequence of reparametrizations v k then satisfies | dv k ( z ) | ≤ z ∈ D ǫ k R k , so by elliptic regularity, v k has a subsequence convergent in C ∞ loc ( C ) to a J -holomorphicmap v ∞ : C → W which is not constant since | dv ∞ (0) | = lim k →∞ | dv k (0) | = 1. Informally, we say thatthe blow-up of the derivatives at z k has caused a plane to “bubble off”. However,(9.2) implies that for every R >
0, one can write ǫ k R k ≥ R for k sufficiently largeand thus Z D R v ∗∞ ω = lim k →∞ Z D R v ∗ k ω ≤ lim k →∞ Z D ǫkRk v ∗ k ω = 0 , implying R C v ∗∞ ω = 0. It follows that v ∞ must be constant, so we have a contradic-tion. (cid:3) ectures on Symplectic Field Theory To obtain the uniform limit of u ( s, · ) as s → ∞ , we now pick any sequence ofnonnegative numbers s k → ∞ and consider the sequence of J -holomorphic half-cylinders u k : [ − s k , ∞ ) × S → W : ( s, t ) u ( s + s k , t ) . By Lemma 9.3, these maps are uniformly C -bounded, so elliptic regularity gives asubsequence converging in C ∞ loc on R × S to a J -holomorphic cylinder u ∞ : R × S → W. Observe that for any c >
0, we can write − s k / ≤ − c for sufficiently large k andthus compute Z [ − c,c ] × S u ∗∞ ω = lim k →∞ Z [ − c,c, ] × S u ∗ k ω ≤ lim k →∞ Z [ − s k / , ∞ ) × S u ∗ k ω = lim k →∞ Z [ s k / , ∞ ) × S u ∗ ω = 0since R Z + u ∗ ω < ∞ . This implies R R × S u ∗∞ ω = 0, so u ∞ is a constant map to somepoint p ∈ W , hence after replacing s k with a subsequence, u ( s k , · ) = u k (0 , · ) → p in C ∞ ( S , W ) as k → ∞ . To finish the proof of (9.1), we need to show that one cannot find two sequences s k → ∞ and s ′ k → ∞ such that u ( s k , · ) → p and u ( s ′ k , · ) → p ′ for distinct points p = p ′ ∈ W . This is an easy consequence of the monotonicity lemma: indeed,if two such sequences exist, then we can find a sequence s ′′ k → ∞ for which theloops u ( s ′′ k , · ) alternate between arbitrarily small neighborhoods of p and p ′ . Since u is continuous, it must then pass through ∂B r ( p ) infinitely many times for r > U k ⊂ Z + such that each u | U k : U k → B r ( q k )is a proper map passing through some point q k ∈ ∂B r ( p ). The monotonicity lemmathen implies Z Z + u ∗ ω ≥ X k Z U k u ∗ ω ≥ X k cr = ∞ , a contradiction. Exercise . Given an area form ω on S = C ∪{∞} and a finite subset Γ ⊂ S ,show that a holomorphic function f : S \ Γ → C has an essential singularity at oneof its punctures if and only if R C f ∗ ω = ∞ . As further preparation for the compactness discussion, we now prove the long-awaited converse of the fact that asymptotically cylindrical curves have finite en-ergy. We work in the setting described in § W, ω ) is a symplectic cobordismwith stable boundary ∂W = − M − ⊔ M + carrying stable Hamiltonian structures Chris Wendl H ± = ( ω ± , λ ± ) with induced hyperplane distributions ξ ± = ker λ ± and Reeb vectorfields R ± . The completion ( c W , ω h ) carries the symplectic structure ω h := d ( h ( r ) λ + ) + ω + on [0 , ∞ ) × M + ω on W ,d ( h ( r ) λ − ) + ω − on ( −∞ , × M − , for some C -small smooth function h ( r ) with h ′ > r = 0,and for a fixed constant r , we define a compact subset W r := ([ − r , × M − ) ∪ M − W ∪ M + ([0 , r ] × M + ) ⊂ c W , outside of which our ω h -tame almost complex structures J ∈ J τ ( ω h , r , H + , H − )are required to be translation-invariant and compatible with H ± . The energy of a J -holomorphic curve u : ( ˙Σ , j ) → ( c W , J ) is defined by E ( u ) := sup f ∈T ( h,r ) Z ˙Σ u ∗ ω f , where T ( h, r ) := (cid:8) f ∈ C ∞ ( R , ( − ǫ, ǫ )) (cid:12)(cid:12) f ′ > f ≡ h near [ − r , r ] (cid:9) . The constant ǫ > J ± ∈J ( H ± ) and X ∈ ξ ± ,(9.3) ( ω ± + κ dλ ± )( X, J ± X ) > X = 0 and κ ∈ ( − ǫ, ǫ ) . This condition implies that every J ∈ J τ ( ω h , r , H + , H − ) is tamed by every ω f forevery f ∈ T ( h, r ), thus all J -holomorphic curves satisfy E ( u ) ≥
0, with equality ifand only if u is constant. Theorem . Assume all closed Reeb orbits in ( M + , H + ) and ( M − , H − ) arenondegenerate, J ∈ J τ ( ω h , r , H + , H − ) , (Σ , j ) is a closed Riemann surface with ˙Σ = Σ \ Γ for some finite subset Γ ⊂ Σ , and u : ( ˙Σ , j ) → ( c W , J ) is a J -holomorphiccurve such that none of the singularities in Γ are removable and E ( u ) < ∞ . Then u is asymptotically cylindrical. Remark . The theorem also holds in the setting of a symplectization ( R × M, J ) with J ∈ J ( H ) for a stable Hamiltonian structure H = ( ω, λ ) on M . Theonly real difference in this case is the slightly simpler definition of energy, E ( u ) = sup f ∈T Z ˙Σ u ∗ ω f , where ω f := d (cid:0) f ( r ) λ (cid:1) + ω and T = (cid:8) f ∈ C ∞ ( R , ( − ǫ, ǫ )) (cid:12)(cid:12) f ′ > (cid:9) . This change necessitates a few trivial modifications to the proof of Theorem 9.6given below. ectures on Symplectic Field Theory
Like removal of singularities, Theorem 9.6 is really a local result, so let us for-mulate a more precise and more general statement in these terms. Let˙ D := D \ { } ⊂ C and define the two biholomorphic maps ϕ + : Z + := [0 , ∞ ) × S → ˙ D : ( s, t ) e − π ( s + it ) ϕ − : Z − := ( −∞ , × S → ˙ D : ( s, t ) e π ( s + it ) . (9.4) Theorem . Suppose J ∈ J τ ( ω h , r , H + , H − ) and u : ˙ D → c W is a J -holomorphic map with E ( u ) < ∞ . Then either the singularity at ∈ D is removableor u is a proper map. In the latter case the puncture is either positive or negative,meaning that u maps neighborhoods of to neighborhoods of {±∞} × M ± , and thepuncture has a well-defined charge , defined as Q = lim ǫ → + Z ∂ D ǫ u ∗ λ ± , which satisfies ± Q > . Moreover, the map ( u R ( s, t ) , u M ( s, t )) := u ◦ ϕ ± ( s, t ) ∈ R × M ± for ( s, t ) ∈ Z ± near infinitysatisfies u R ( s, · ) − T s → c in C ∞ ( S ) as s → ±∞ for T := | Q | and a constant c ∈ R , while for every sequence s k → ±∞ , one canrestrict to a subsequence such that u M ( s k , · ) → γ ( T · ) in C ∞ ( S , M ± ) as k → ∞ for some T -periodic Reeb orbit γ : R /T Z → M ± . If γ is nondegenerate or Morse-Bott, then in fact u M ( s, · ) → γ ( T · ) in C ∞ ( S , M ± ) as s → ±∞ We will not prove this result in its full strength, as in particular the last step(when γ is nondegenerate or Morse-Bott) requires some asymptotic elliptic regularityresults that we do not have space to explain here. Note however that most of theabove statement does not require any nondegeneracy assumption at all. The pricefor this level of generality is that if s k , s ′ k → ±∞ are two distinct sequences, thenwe have no guarantee in general that the two Reeb orbits obtained as limits ofsubsequences of u M ( s k , · ) and u M ( s ′ k , · ) will be the same; at present, neither anexample of this rather unpleasant possibility nor any general argument to rule itout is known. If one of these orbits is assumed to be isolated, however—which isalways true when the Reeb vector field is nondegenerate—then we will be able toshow that both are the same up to parametrization, hence geometrically , u M ( s, t )lies in arbitrarily small neighborhoods of the orbit γ as s → ±∞ . This turns out tobe also true in the more general Morse-Bott setting, though it is then much harderto prove since γ need not be isolated. Once u M ( s, · ) is localized near γ , one canuse the nondegeneracy condition as we did in the Fredholm theory of Lecture 4 todevelop asymptotic regularity results that give much finer control over the behaviorof u M as s → ±∞ , implying in particular that u M ( s, · ) → γ ( T · ) in C ∞ ( S , M ± ). Chris Wendl
For details on this step, we refer to the original sources: [
HWZ96 , HWZ01 ] for thenondegenerate case, and [
HWZ96 , Bou02 ] when the Reeb vector field is Morse-Bott. Those papers deal exclusively with the contact case, but the setting of generalstable Hamiltonian structures is also dealt with in [
Sie08 ].Ignoring the final step for now, the proof of Theorem 9.8 will reuse most of thetechniques that we already saw in our proof of removal of singularities in § u has a removable singularity, it is a proper map, and for anysequence s k → ±∞ , the holomorphic half-cylinders defined by u k ( s, t ) = u ◦ ϕ ± ( s + s k , t )on a sequence of increasingly large half-cylinders must have a subsequence convergingin C ∞ loc ( R × S ) to either a constant map or a trivial cylinder. The first case will turnout to mean (as in Theorem 9.2) that the puncture is removable, and the secondimplies asymptotic convergence to a closed Reeb orbit.One major difference between the proof of Theorem 9.8 and removal of singu-larities is that since c W is noncompact, sequences of curves in c W with uniformlybounded first derivatives need not be locally C -bounded. This issue will arise bothin the bubbling argument to prove | du k ( s, t ) | ≤ C and in the analysis of the sequence u k itself. In such cases, one can use the R -translation action(9.5) τ c : R × M ± → R × M ± : ( r, x ) ( r + c, x ) for c ∈ R on suitable subsets of the cylindrical ends to replace unbounded sequences withuniformly C -bounded sequences of curves mapping into R × M + or R × M − . These R -translations are the reason why our definition of energy needs to be somethingslightly more complicated than just the symplectic area R ˙Σ u ∗ Ω for a single choiceof symplectic form. To understand bubbling in the presence of arbitrarily large R -translations, we will need the following lemma. Lemma . Suppose J ∈ J ( H ) for some stable Hamiltonian structure H =( ω, λ ) on an odd-dimensional manifold M , and u : ( ˙Σ , j ) → ( R × M, J ) is a J -holomorphic curve satisfying E ( u ) < ∞ and Z ˙Σ u ∗ ω = 0 . If ˙Σ = C , then u is constant. If ˙Σ = R × S , then u either is constant or isbiholomorphically equivalent to a trivial cylinder over a closed Reeb orbit. Proof.
Denote ξ = ker λ and let π ξ : T ( R × M ) → ξ denote the projection along the subbundle spanned by ∂ r (the unit vector field inthe R -direction) and the Reeb vector field R . Then since ω annihilates both ∂ r and R , for any local holomorphic coordinates ( s, t ) on a subset of ˙Σ, the compatibilityof J | ξ with ω | ξ implies u ∗ ω ( ∂ s , ∂ t ) = ω ( ∂ s u, ∂ t u ) = ω ( ∂ s u, J ∂ s u ) = ω ( π ξ ∂ s u, J π ξ ∂ s u ) ≥ , ectures on Symplectic Field Theory hence R ˙Σ u ∗ ω ≥ J -holomorphic curve, and equality means that u iseverywhere tangent to the subbundle spanned by ∂ r and R . This implies that im u is contained in the image of some J -holomorphic plane of the form u γ : C → R × M : s + it ( s, γ ( t )) , where γ : R → M is a (not necessarily periodic) orbit of R . If γ is not periodic,then u γ is embedded, hence there exists a unique (and necessarily holomorphic) mapΦ : ( ˙Σ , j ) → ( C , i ) such that u = u γ ◦ Φ. If on the other hand γ is periodic withminimal period T >
0, then u γ descends to an embedding of the cylinderˆ u γ : C /iT Z → R × M, and we can view u γ as a covering map to this embedded cylinder. Now there existsa unique holomorphic map Φ : ˙Σ → C /iT Z such that u = ˆ u γ ◦ Φ. If ˙Σ = C , thensince π ( C ) = 0 implies that Φ can be lifted to a (necessarily holomorphic) map e Φ : C → C with u γ ◦ e Φ = u . Relabeling symbols, we conclude that in general if˙Σ = C , then u = u γ ◦ Φ for a holomorphic map Φ : C → C .Let us consider all cases in which the factorzation u = u γ ◦ Φ exists, whereΦ : ( ˙Σ , j ) → ( C , i ) is holomorphic and ˙Σ = Σ \ Γ for a closed Riemann surface(Σ , j ). We will now use the removable singularity theorem for Φ : ˙Σ → S \ { } toshow that unless Φ is constant, R ˙Σ u ∗ ω f = ∞ for suitable choices of f ∈ T . Thisintegral can be rewritten as(9.6) Z ˙Σ u ∗ ω f = Z ˙Σ Φ ∗ u ∗ γ ω f = Z ˙Σ Φ ∗ d ( f ( s ) dt ) = Z ˙Σ Φ ∗ ( f ′ ( s ) ds ∧ dt )since ω f = d (cid:0) f ( r ) λ (cid:1) + ω and u γ ( s, t ) = ( s, γ ( t )). Since f ′ > f ′ ( s ) ds ∧ dt is an areaform on C with infinite area. We claim now that for suitable choices of f ∈ T , onecan find an area form Ω on S = C ∪ {∞} such that Ω ≤ f ′ ( s ) ds ∧ dt . To see this,let us change coordinates so that ∞ becomes 0: setting Ψ : C ∗ → C ∗ : z /z , aslightly tedious but straightforward computation givesΨ ∗ ( f ′ ( s ) ds ∧ dt ) = f ′ ( s/ | z | ) 1 | z | (cid:18) st ) | z | (cid:19) ds ∧ dt ≥ f ′ ( s/ | z | ) 1 | z | ds ∧ dt for z = s + it ∈ C \ { } . (9.7)We need to show that this 2-form can be bounded away from 0 as z →
0. Let uschoose f ∈ T such that(9.8) f ( r ) = ± (cid:16) ǫ − ǫ r (cid:17) for ± r ≥ f arbitrarily to [ − ,
1] such that f ′ >
0. We can then find a constant c > f ′ satisfies f ′ ( r ) > min n c, ǫ r o for all r ∈ R . Chris Wendl
Plugging this into (9.7) givesΨ ∗ ( f ′ ( s ) ds ∧ dt ) ≥ min (cid:26) c | z | , ǫ s (cid:27) ds ∧ dt, which clearly blows up as | z | →
0. With this established, we observe that for anynumber
C >
0, the fact that f ′ ( s ) ds ∧ dt has infinite area implies we can choose anarea form Ω on S withΩ ≤ f ′ ( s ) ds ∧ dt on S \ {∞} and Z S Ω > C. We now have two possibilities:(1) If R ˙Σ Φ ∗ Ω < ∞ , then Theorem 9.2 implies that the singularities of Φ : ˙Σ → C at Γ are all removable, i.e. Φ extends to a holomorphic map (Σ , j ) → ( S , i ), which has a well-defined mapping degree k ≥
0. Then Z ˙Σ u ∗ ω f = Z ˙Σ Φ ∗ ( f ′ ( s ) ds ∧ dt ) ≥ Z ˙Σ Φ ∗ Ω = Z Σ Φ ∗ Ω = k Z S Ω > kC. Since
C > R ˙Σ u ∗ ω f = ∞ unless k = 0, meaning Φ is constant.(2) If R ˙Σ Φ ∗ Ω = ∞ (meaning there is an essential singularity, cf. Exercise 9.5),then since Φ ∗ ( f ′ ( s ) ds ∧ dt ) ≥ Φ ∗ Ω, (9.6) implies R C u ∗ ω f = ∞ .Since u is constant whenever Φ is, this completes the proof for ˙Σ = C .If ˙Σ = R × S , then it remains to deal with the case where the factorization u = u γ ◦ Φ does not exist because γ is periodic. If the minimal period is T >
0, thenlet us in this case redefine u γ as an embedded J -holomorphic trivial cylinder u γ : R × S : ( s, t ) ( T s, γ ( T t )) . Since the new u γ is embedded, we can now write u = u γ ◦ Φ for a unique holomorphicmap Φ : R × S → R × S . Identifying R × S biholomorphically with S \ { , ∞} ,we claim that Φ extends to a holomorphic map S → S . Indeed, by the removablesingularity theorem, this is true if and only if R R × S Φ ∗ Ω < ∞ for some area formΩ on S . Notice that u ∗ γ ω f = T · f ′ ( T s ) ds ∧ dt , defines an area form on R × S with finite area for any f ∈ T since R ∞−∞ f ′ ( s ) ds < ∞ ; this is equivalent to theobservation that trivial cylinders always have finite energy. Using the biholomorphicmap ( s, t ) e π ( s + it ) to identify R × S with C ∗ = S \ { , ∞} and using coordinates z = x + iy on the latter, another tedious but straightforward computation gives u ∗ γ ω f = T π f ′ (cid:0) T π log | z | (cid:1) | z | dx ∧ dy for z = x + iy ∈ C ∗ . Now suppose f ∈ T is chosen as in (9.8). Then one can check that the positivefunction in front of dx ∧ dy in the above formula goes to + ∞ as | z | →
0; this meansthat one can find an area form Ω on C with Ω ≤ u ∗ γ ω f on C ∗ . The singularity at+ ∞ ∈ S can be handled in a similar way, thus we can find an area form Ω on S ectures on Symplectic Field Theory such that Ω ≤ u ∗ γ ω f on R × S . Now since E ( u ) < ∞ , we have Z R × S Φ ∗ Ω ≤ Z R × S Φ ∗ u ∗ γ ω f = Z R × S u ∗ ω f < ∞ , so by Theorem 9.2, Φ has a holomorphic extension S → S , which is then a mapof degree k ≥ − ( { , ∞} ) ⊂ { , ∞} . If k = 0 then Φ is constant, and sois u . Otherwise, Φ is surjective and thus hits both 0 and ∞ , but it can only do thisat either 0 or ∞ , thus it either fixes both or interchanges them. After composingwith a biholomorphic map of S preserving R × S , we may assume without loss ofgenerality that Φ(0) = 0 and Φ( ∞ ) = ∞ . This makes Φ a polynomial with only onezero, hence as a map on C ∪ {∞} , Φ( z ) = cz k for some c ∈ C ∗ . Up to biholomorphicequivalence, Φ( z ) is then z k , which appears in cylindrical coordinates as the map( s, t ) ( ks, kt ), so u is now the trivial cylinder u ( s, t ) = u γ ( ks, kt ) = ( kT s, γ ( kT t ))over the k -fold cover of γ . (cid:3) Remark . It may be useful for some applications to observe that Lemma 9.9does not require M to be compact. In contrast, the compactness arguments in thislecture almost always depend on the assumption that W and M ± are compact—without this, one would need add some explicit assumption to guarantee local C -bounds on sequences of holomorphic curves, e.g. the assumption in Theorem 9.2that u ( D \ { } ) is contained in a compact subset.Before continuing, it is worth noting that neither of the two definitions of energystated above (one for curves in c W and the other for symplectizations) is unique,i.e. each can be tweaked in various ways such that the results of this section stillhold. Indeed, the original definitions appearing in [ Hof93 , BEH + ] are slightlydifferent, but equivalent to these. The next lemma illustrates one further exampleof this freedom, which will be useful in some of the arguments below. Lemma . Given a stable Hamiltonian structure H = ( ω, λ ) on M , a suffi-ciently small constant ǫ > as in (9.3) , and J ∈ J ( H ) , consider the alternativenotion of energy for J -holomorphic curves u : ( ˙Σ , j ) → ( R × M, J ) defined by E ( u ) = sup f ∈T Z ˙Σ u ∗ ω f where ω f = d ( f ( r ) λ ) + ω and T = (cid:8) f ∈ C ∞ ( R , ( a, b )) (cid:12)(cid:12) f ′ > (cid:9) for some constants − ǫ ≤ a < b ≤ ǫ . Then if E ( u ) denotes the energy as written inRemark 9.7, there exists a constant c > , depending on the data a , b , ǫ and H butnot on u , such that cE ( u ) ≤ E ( u ) ≤ E ( u ) . Chris Wendl
Proof.
The second of the two inequalities is immediate since T ⊂ T . For thefirst inequality, note that since ǫ > c > X ∈ T ( R × M ) and every κ ∈ [ − ǫ, ǫ ],(9.9) 1 c ( ω + κ dλ )( X, J X ) ≤ ω ( X, J X ) ≤ c ( ω + κ dλ )( X, J X ) . This uses (9.3) and the fact that dλ annihilates ker ω . Now suppose f ∈ T , choosea constant δ ∈ (0 , b − a ] and define ˜ f ∈ T by˜ f ( r ) = δ ǫ f ( r ) + a + b . Then ˜ f ′ ( r ) = δ ǫ f ′ ( r ), and given a J -holomorphic curve u : ˙Σ → R × M , we canwrite ω f = ω + f ( r ) dλ + f ′ ( r ) dr ∧ λ and use (9.9) to estimate Z ˙Σ u ∗ ω f = Z ˙Σ u ∗ ( ω + f ( r ) dλ ) + Z ˙Σ u ∗ ( f ′ ( r ) dr ∧ λ ) ≤ c Z ˙Σ u ∗ ω + 2 ǫδ Z ˙Σ u ∗ (cid:16) ˜ f ′ ( r ) dr ∧ λ (cid:17) ≤ c Z ˙Σ u ∗ (cid:16) ω + ˜ f ( r ) dλ (cid:17) + 2 ǫδ Z ˙Σ u ∗ (cid:16) ˜ f ′ ( r ) dr ∧ λ (cid:17) . If c ≥ ǫb − a , then we can choose δ := 2 ǫ/c ≤ b − a and rewrite the last expression as c Z ˙Σ u ∗ (cid:16) ω + ˜ f ( r ) dλ (cid:17) + 2 ǫδ Z ˙Σ u ∗ (cid:16) ˜ f ′ ( r ) dr ∧ λ (cid:17) = c Z ˙Σ u ∗ (cid:16) ω + ˜ f ( r ) dλ + ˜ f ′ ( r ) dr ∧ λ (cid:17) = c Z ˙Σ u ∗ ω ˜ f ≤ c E ( u ) . On the other hand if c < ǫb − a , we can set δ := b − a and write c Z ˙Σ u ∗ (cid:16) ω + ˜ f ( r ) dλ (cid:17) + 2 ǫδ Z ˙Σ u ∗ (cid:16) ˜ f ′ ( r ) dr ∧ λ (cid:17) ≤ ǫb − a Z ˙Σ u ∗ (cid:16) ω + ˜ f ( r ) dλ + ˜ f ′ ( r ) dr ∧ λ (cid:17) = 2 ǫb − a Z ˙Σ u ∗ ω ˜ f ≤ ǫb − a E ( u ) . (cid:3) With this preparation out of the way, we now begin in earnest with the proofof Theorem 9.8. Assume u : ˙ D → c W is a J -holomorphic punctured disk satisfying E ( u ) < ∞ . Using the maps ϕ ± : Z ± → ˙ D defined in (9.4), we shall write u ± := u ◦ ϕ ± : Z ± → c W and observe that these reparametrizations have no impact on the energy, i.e. E ( u ± ) = sup f ∈T ( h,r ) Z Z ± ( u ◦ ϕ ± ) ∗ ω f = sup f ∈T ( h,r ) Z ˙ D u ∗ ω f = E ( u ) . ectures on Symplectic Field Theory Fix a Riemannian metric on c W that is translation-invariant on the cylindrical ends,and fix the standard metric on the half-cylinders Z ± . We will use these metricsimplicitly whenever referring to quantities such as | du ± ( z ) | . Lemma . There exists a constant
C > such that | du + ( s, t ) | ≤ C for all ( s, t ) ∈ Z + . Proof.
We use a bubbling argument as in the proof of Lemma 9.3. Suppose thecontrary, so there exists a sequence z k = ( s k , t k ) ∈ Z + with R k := | du + ( z k ) | → ∞ .Choose a sequence ǫ k > ǫ k → ǫ k R k → ∞ , and using Lemma 9.4,assume without loss of generality that | du + ( z ) | ≤ R k for all z ∈ D ǫ k ( z k ) . Define a rescaled sequence of J -holomorphic disks by v k : D ǫ k R k → c W : z u ◦ ϕ + ( z k + z/R k ) . These satisfy | dv k | ≤ C -boundedsince their images may escape to infinity. We distinguish three possibilities, at leastone of which must hold: Case 1: v k (0) has a bounded subsequence. Then the corresponding subsequence of v k : D ǫ k R k → c W is uniformly C -boundedon every compact subset and thus (by elliptic regularity) has a further subsequenceconvergent in C ∞ loc ( C ) to a J -holomorphic plane v ∞ : C → c W with | dv ∞ (0) | = lim k →∞ | dv k (0) | = 1. But by the same argument we used in theproof of Lemma 9.3, the fact that R Z + u ∗ + ω f < ∞ for any choice of f ∈ T ( h, r )implies Z C v ∗∞ ω f = 0 , hence v ∞ is constant, and this is a contradiction. Case 2: v k (0) has a subsequence diverging to { + ∞} × M + .Restricting to this subsequence, suppose v k (0) ∈ { r k } × M + , so r k → ∞ , and assume without loss of generality that r k > r for all k . Let˜ R k ∈ (0 , ǫ k R k ] for each k denote the largest radius such that v k ( D ˜ R k ) ⊂ ( r , ∞ ) × M + .Then ˜ R k → ∞ since | dv k | is bounded. Now using the R -translation maps τ r definedin (9.5), define ˜ v k := τ − r k ◦ v k | D ˜ Rk : D ˜ R k → R × M + . Since we’re using a translation-invariant metric on [ r , ∞ ) × M + , ˜ v k is now a uni-formly C -bounded sequence of maps into R × M + . Elliptic regularity thus providesa subsequence convergent in C ∞ loc ( C ) to a plane v ∞ : C → R × M + , Chris Wendl which is J + -holomorphic, where J + ∈ J ( H + ) denotes the restriction of J to [ r , ∞ ) × M + , extended over R × M + by R -invariance. We claim,(9.10) E ( v ∞ ) < ∞ and Z C v ∗∞ ω + = 0 , where E ( v ∞ ) is now defined as in Remark 9.7. By Lemma 9.11, the first part of theclaim will follow if we can fix a constant a ∈ ( − ǫ, ǫ ) and establish a uniform bound Z C v ∗∞ Ω + f ≤ C, with Ω + f := ω + + d (cid:0) f ( r ) λ + (cid:1) , for all smooth and strictly increasing functions f : R → ( a, ǫ ). For convenience in the following, we shall assume a > h ( r ). Now if f is such a function, then for any R > Z D R v ∗∞ Ω + f = lim k →∞ Z D R v ∗ k τ ∗− r k Ω + f = lim k →∞ Z D R v ∗ k Ω + f k , where f k ( r ) := f ( r − r k ). Notice that the dependence of the last integral on f k islimited to the interval ( r , ∞ ) since v k ( D R ) ⊂ ( r , ∞ ) × M + . Then since f > a >h ( r ) by assumption, there exists for each k a function h k ∈ T ( h, r ) that matches f k outside some neighborhood of ( −∞ , r ] and thus satisfies Z D R v ∗ k Ω + f k = Z D R v ∗ k ω h k ≤ Z D ǫkRk v ∗ k ω h k = Z D ǫk ( z k ) u ∗ + ω h k ≤ Z Z + u ∗ + ω h k ≤ E ( u ) . This is true for every
R >
R > f ∈ T ( h, r ). Observe that since we canassume (after perhaps passing to a subsequence) the disks D ǫ k ( z k ) are all disjoint,0 = lim k →∞ Z D ǫk ( z k ) u ∗ + ω f = lim k →∞ Z D ǫkRk v ∗ k ω f = lim k →∞ Z D ǫkRk ˜ v ∗ k τ ∗ r k ω f ≥ lim k →∞ Z D R ˜ v ∗ k τ ∗ r k ω f = lim k →∞ Z D R ˜ v ∗ k Ω + f k , where now f k ( r ) := f ( r + r k ). Writing Ω + f k = ω + + d (cid:0) f k ( r ) λ + (cid:1) = ω + + f k ( r ) dλ + + f ′ k ( r ) dr ∧ λ + , we can choose f such that f ′ ( r ) = f ′ ( r + r k ) → k → ∞ ,so the third term contributes nothing to the integral. For the second term, let f + := lim k →∞ f k ( r ) = lim r →∞ f ( r ), so the calculation above becomes0 ≥ Z D R v ∗∞ ( ω + + f + dλ + ) . Now observe that since f + ∈ [ − ǫ, ǫ ], condition (9.3) implies that the 2-form ω + + f + dλ + is nondegenerate on ξ + , and it also annihilates ∂ r and R + , so the vanishingof this integral implies that v ∞ is everywhere tangent to ∂ r and R + over D R . But R > R C v ∗∞ ω + = 0. With the claim established, we apply Lemma 9.9 and conclude that v ∞ is constant, contradicting the fact that | dv ∞ (0) | = 1. Case 3: v k (0) has a subsequence diverging to {−∞} × M − .This is simply the mirror image of case 2: writing the restriction of J to ( −∞ , − r ] × ectures on Symplectic Field Theory M − as J − , one can follow the same bubbling argument but translate up and insteadof down, giving rise to a limiting nonconstant J − -holomorphic plane v ∞ : C → R × M − that has finite energy but R C v ∗∞ ω − = 0, in contradiction to Lemma 9.9. (cid:3) Consider now a sequence s k → ∞ and construct the J -holomorphic half-cylinders u k : [ − s k , ∞ ) × S → c W : ( s, t ) u + ( s + s k , t ) . The derivatives | du k | are uniformly bounded due to Lemma 9.12, though again, u k might fail to be uniformly bounded in C . We distinguish three cases. Case 1: u k (0 , has a bounded subsequence. Then the corresponding subsequence of u k is uniformly C -bounded on compactsubsets and thus has a further subsequence converging in C ∞ loc ( R × S ) to a J -holomorphic cylinder u ∞ : R × S → c W .
For any f ∈ T ( h, r ) and any c >
0, we have Z [ − c,c ] × S u ∗∞ ω f = lim k →∞ Z [ − c,c ] × S u ∗ k ω f ≤ lim k →∞ Z [ − s k / , ∞ ) × S u ∗ k ω f = lim k →∞ Z [ s k / , ∞ ) × S u ∗ + ω f = 0(9.11)since R Z + u ∗ + ω f < ∞ . It follows that R R × S u ∗∞ ω f = 0, so u ∞ is a constant map tosome point p ∈ c W , implying that after passing to a subsequence of s k , u + ( s k , · ) → p in C ∞ ( S , c W ) as k → ∞ . Case 2: u k (0 , has a subsequence diverging to { + ∞} × M + . Passing to the corresponding subsequence of u k , suppose u k (0 , ∈ { r k } × M + , so r k → ∞ . Since the derivatives | du k | are uniformly bounded, we can then find asequence of intervals [ − R − k , R + k ] ⊂ [ − s k , ∞ ) such that u k ([ − R − k , R + k ] × S ) ⊂ [ r , ∞ ) × M + and R ± k → ∞ . Now the translated sequence τ − r k ◦ u k | [ − R − k ,R + k ] × S : [ − R − k , R + k ] × S → R × M + is uniformly C -bounded on compact subsets and thus has a subsequence covergingin C ∞ loc to a J + -holomorphic cylinder u ∞ : R × S → R × M + , where J + again denotes the restriction of J to [ r , ∞ ) × M + , extended over R × M + by R -translation. We claim that this cylinder satisfies E ( u ∞ ) < ∞ and Z R × S u ∗∞ ω + = 0 . Chris Wendl
The proof of this should be an easy exercise if you understood the proofs of (9.10)and (9.11) above, so I will leave it as such. Lemma 9.9 now implies that u ∞ is eitherconstant or is a reparametrization of a trivial cylinder u γ : R × S → R × M + : ( s, t ) ( T s, γ ( T t ))for some Reeb orbit γ : R /T Z → M + with period T >
0. More precisely, all thebiholomorphic reparametrizations of R × S are of the form ( s, t ) ( ± s + a, ± t + b ), thus after shifting the parametrization of γ , we can write u ∞ without loss ofgenerality in the form(9.12) u ∞ ( s, t ) = ( ± T s + a, γ ( ± T t ))for some constant a ∈ R and a choice of signs to be determined below (see Lemma 9.16). Case 3: u k (0 , has a subsequence diverging to {−∞} × M − . Writing J − := J | ( −∞ , − r ] × M − ∈ J ( H − ) and imitating the argument for case 2, wesuppose u k (0 , ∈ {− r k } × M − with r k → ∞ and obtain a subsequence for which τ r k ◦ u k converges in C ∞ loc ( R × S ) to a J − -holomorphic cylinder u ∞ : R × S → R × M − , where u ∞ is either a constant or takes the form (9.12) for some orbit Reeb γ : R /T Z → M − of period T > c W to define a metric dist C ( · , · ) on the space of continuous loops S → c W . Lemma . Given δ > , there exists s ≥ such that for every s ≥ s , theloop u + ( s, · ) : S → c W satisfies dist C ( u + ( s, · ) , ℓ ) < δ, where ℓ : S → c W either is constant or is a loop of the form ℓ ( t ) = ( r, γ ( ± T t )) in [ r , ∞ ) × M + or ( −∞ , r ] × M − for some constant r ∈ R and Reeb orbit γ : R /T Z → M ± of period T > . Proof.
If not, then there exists a sequence s k → ∞ such that each of the loops u + ( s k , · ) lies at C -distance at least δ away from any loop of the above form. How-ever, the preceding discussion then gives a subsequence for which u ( s k , · ) becomesarbitrarily C ∞ -close to such a loop, so this is a contradiction. (cid:3) Lemma . If u : ˙ D → c W is not bounded, then it is proper. Proof.
We use the monotonicity lemma. Suppose there exists a sequence( s k , t k ) ∈ Z + such that u + ( s k , t k ) diverges to { + ∞} × M + . This implies s k → ∞ ,and we claim then that for every R ≥ r , there exists s ≥ u + (( s , ∞ ) × S ) ⊂ ( R, ∞ ) × M + . If not, then we find R ≥ r and a sequence ( s ′ k , t ′ k ) ∈ Z + with s ′ k → ∞ suchthat u + ( s ′ k , t ′ k ) ( R, ∞ ) × M + for every k . By continuity, we are free to suppose u + ( s ′ k , t ′ k ) ∈ { R }× M + for all k since Lemma 9.13 implies u + ( { s k }× S ) ⊂ (2 R, ∞ ) × M + for k sufficiently large. Using Lemma 9.13 again, we also have u + ( { s ′ k } × S ) ⊂ ( R − , R + 1) × M + ectures on Symplectic Field Theory for all k large. Assuming 2 R > R + 2 without loss of generality, we can thereforefind infinitely many pairwise disjoint annuli of the form [ s ′ k , s j ] × S ⊂ Z + containingopen sets that u maps properly to small balls centered at points in { R + 2 } × M + .Choosing any f ∈ T ( h, r ), the monotonicity lemma implies that each of thesecontributes at least some fixed amount to R Z + u ∗ + ω f , contradicting the assumptionthat E ( u ) < ∞ . A similar argument works if u + ( s k , t k ) diverges to {−∞} × M − , proving that forevery R ≥ r , there exists s ≥ u + (( s , ∞ ) × S ) ⊂ ( −∞ , − R ) × M − . (cid:3) If u is bounded, then the singularity at 0 is removable by Theorem 9.2. If not,then Lemma 9.14 implies that it maps neighborhoods of the puncture to neighbor-hoods of either { + ∞} × M + or {−∞} × M − , and we shall refer to the puncture as positive or negative accordingly. Lemma . If the puncture is positive/negative, then the limit Q := lim s →∞ Z S u + ( s, · ) ∗ λ ± ∈ R exists. Proof.
If the puncture is positive, fix s ≥ u + ([ s , ∞ ) × S ) ⊂ [ r , ∞ ) × M + . Then by Stokes’ theorem, it suffices to show that the integral R [ s , ∞ ) × S u ∗ + dλ + exists, which is true if(9.13) Z [ s , ∞ ) × S (cid:12)(cid:12) u ∗ + dλ + (cid:12)(cid:12) < ∞ . We claim first that R [ s , ∞ ) × S u ∗ + ω + < ∞ . Indeed, for any s > s and f ∈ T ( h, r ),we have E ( u ) ≥ Z [ s ,s ] × S u ∗ + ω f = Z [ s ,s ] × S u ∗ + ω + + Z [ s ,s ] × S u ∗ + d ( f ( r ) λ + ) . Applying Stokes’ theorem, the second term becomes the sum of some number notdependent on s and the integral Z S u + ( s, · ) ∗ ( f ( r ) λ + ) = Z S [ f ◦ u + ( s, · )] u + ( s, · ) ∗ λ + , which is bounded as s → ∞ since f and | du + | are both bounded. This proves that R [ s ,s ] × S u ∗ + ω + is also bounded as s → ∞ , and since u ∗ + ω + ≥
0, the claim follows.Now observe that since dλ + annihilates the kernel of ω + and the latter tames J on ξ + , there exists a constant c > | u ∗ + dλ + | ≤ c | u ∗ + ω + | , implying (9.13).An analogous argument works if the puncture is negative. (cid:3) The fact that c W is noncompact is not a problem for this application of the monotonicitylemma, as we are only using it in the compact subset W R ⊂ c W . Chris Wendl
The number Q ∈ R defined in the above lemma matches what we referred to inthe statement of Theorem 9.8 as the charge of the puncture. Lemma . If the puncture is nonremovable and Q = 0 , then the puncture ispositive/negative if and only if Q > or Q < respectively. In either case, given anysequence s k → ∞ with u + ( s k , ∈ {± r k } × M ± , one can find a sequence R k ∈ [0 , s k ] with R k → ∞ such that u + maps [ s k − R k , ∞ ) × S into the positive/negativecylindrical end for every k , and the sequence of half-cylinders u k : [ − R k , ∞ ) × S → R × M + or u k : ( −∞ , R k ] × S → R × M − defined by u k ( s, t ) = τ ∓ r k ◦ u ± ( s ± s k , t ) has a subsequence convergent in C ∞ loc ( R × S ) to a J ± -holomorphic cylinder of the form u ∞ : R × S → R × M ± : ( s, t ) ( T s + a, γ ( T t )) for some constant a ∈ R and Reeb orbit γ : R /T Z → M ± with period T := ± Q . Proof.
Assume the puncture is either positive or negative and Q = 0. In thediscussion preceding Lemma 9.13, we showed that the sequence u ′ ( s, t ) := τ ∓ r k ◦ u + ( s + s k , t ) defined on [ − R k , ∞ ) × S has a subsequence convergent in C ∞ loc to a J ± -holomorphic cylinder u ′∞ : R × S → R × M ± which is either constant or of theform(9.14) u ′∞ ( s, t ) = ( σT s + a, γ ( σT t ))for some a ∈ R , σ = ± γ : R /T Z → M ± of period T >
0. Wethen have0 = Q = lim s →∞ Z S u + ( s, · ) ∗ λ ± = lim k →∞ Z S u ′ k (0 , · ) ∗ λ ± = Z S u ′∞ (0 , · ) ∗ λ ± , so u ′∞ cannot be constant, and from (9.14) we deduce Q = σT , hence u ′∞ ( s, t ) =( Qs + a, γ ( Qt )). Writing u + ( s, t ) = ( u R ( s, t ) , u M ( s, t )) ∈ R × M ± for s sufficientlylarge, it follows that every sequence s k → ∞ admits a subsequence for which ∂ s u R ( s k , · ) → Q in C ∞ ( S , R ) , and consequently ∂ s u R ( s, · ) → Q in C ∞ ( S , R ) as s → ∞ . This proves that the signof Q matches the sign of the puncture whenever Q = 0. The stated formula for u ∞ now follows by adjusting all the appropriate signs in the case Q < (cid:3) Lemma . If the puncture is nonremovable, then Q = 0 . Proof.
Assume on the contrary that u is a proper map, say with a positivepuncture, but Q = 0. In this case, the argument of the previous lemma shows thatthe limiting map u ∞ : R × S → R × M + will always be constant , thus for everysequence s k → ∞ , there exists a point p ∈ M + such that u + ( s k , ∈ { r k } × M + with r k → ∞ and τ − r k ◦ u + ( s k , · ) → (0 , p ) ∈ R × M + in C ∞ ( S , R × M + ) as k → ∞ . In particular, this implies that all derivatives of u + decay to 0 as s → ∞ . Intuitively,this should suggest to you that portions of u + near infinity will have improbablysmall symplectic area, perhaps violating the monotonicity lemma—this will turn out ectures on Symplectic Field Theory to be true, but we have to be a bit clever with our argument since u + is unbounded.We will make this argument precise by translating pieces of u + downward so thatwe only compute its symplectic area in [0 , × M + . Fix a function f : R → ( − ǫ, ǫ )with f ′ > + f = ω + + d ( f ( r ) λ + ).Given a small number δ >
0, we can find s ≥ | du + ( s, t ) | < δ for all s ≥ s and each of the loops u + ( s, · ) for s ≥ s is δ -close to a constant in C ( S ).Assume u + ( s , ∈ { R }× M + and choose s > s such that u + ( s , ∈ { R +2 }× M + ,which is possible since u + ( s, t ) → { + ∞} × M + as s → ∞ . Now consider the J + -holomorphic annulus v δ := τ − R ◦ u + | [ s ,s ] × S : [ s , s ] × S → R × M + . We claim that R [ s ,s ] × S v ∗ δ Ω + f can be made arbitrarily small by choosing δ suitablysmall. Indeed, we can use Stokes’ theorem to write this integral as Z [ s ,s ] × S v ∗ δ Ω + f = Z [ s ,s ] × S v ∗ δ ω + + Z [ s ,s ] × S v ∗ δ d ( f ( r ) λ + )= Z [ s ,s ] × S v ∗ δ ω + + Z S [ v δ ( s , · ) ∗ ( f ( r ) λ + ) − v δ ( s , · ) ∗ ( f ( r ) λ + )] . The second term is small because f ( r ) is bounded and | v δ ( s, · ) ∗ λ + | is small in pro-portion to | dv δ ( s, t ) | = | du + ( s, t ) | for s ≥ s . For the first term, observe that sinceboth of the loops v δ ( s i , · ) for i = 0 , v i : D → R × M + for which (cid:12)(cid:12)R D ¯ v ∗ i ω + (cid:12)(cid:12) may be assumedarbitrarily small. Moreover, since all of the loops v δ ( s, · ) are similarly contractible,the union of these two disks with the annulus v δ defines a closed cycle in M + that istrivial in H ( M + ), hence the integral of the closed 2-form ω + over this cycle vanishes,implying Z [ s ,s ] × S v ∗ δ ω + = Z D ¯ v ∗ ω + − Z D ¯ v ∗ ω + , which is therefore arbitrarily small, and this proves the claim.To finish, notice that since v δ maps its boundary components to small neighbor-hoods of { } × M + and { } × M + , one can fix a suitable choice of radius r > v δ must pass through a point in p ∈ { } × M + for which the boundary of v δ is outside the ball B r ( p ). The monotonicity lemma then bounds the symplecticarea of v δ from below by a constant times r , but since we can also make this areaarbitrarily small by choosing δ smaller, this is a contradiction.As usual, the case of a negative puncture can be handled similarly. (cid:3) We’ve now proved every statement in Theorem 9.8 up to the final detail aboutthe case where the asymptotic orbit is nondegenerate or Morse-Bott. The com-plete proof of this part requires delicate analytical results from [
HWZ96 , HWZ01 , HWZ96 , Bou02 ], but we can explain the first step for the nondegenerate case.In the following, we say that a closed Reeb orbit γ : R /T Z → M ± is isolated if,after rescaling the domain to write it as an element of C ∞ ( S , M ± ), there existsa neighborhood γ ∈ U ⊂ C ∞ ( S , M ± ) such that all closed Reeb orbits in U arereparametrizations of γ . Chris Wendl
Lemma . Suppose the puncture is nonremovable, write u + ( s, t ) = ( u R ( s, t ) , u M ( s, t )) ∈ R × M ± for s ≥ sufficiently large, and suppose s k → ∞ is a sequence and γ : R /T Z → M ± is a Reeb orbit such that u M ( s k , · ) → γ ( T · ) in C ∞ ( S , M ± ) . If γ is isolated, then for every neighborhood U ⊂ C ∞ ( S , M ± ) of the set of parametriza-tions { γ ( · + θ ) | θ ∈ S } , we have u M ( s, · ) ∈ U for all sufficiently large s . Proof.
Note first that if γ is isolated, then its image admits a neighborhoodim γ ⊂ V ⊂ M ± such that no point in V \ im γ is contained in another Reeborbit of period T . Indeed, we could otherwise find a sequence of T -periodic Reeborbits passing through a sequence of points in V \ im γ that converge to a pointin im γ . Since their derivatives are determined by the Reeb vector field and aretherefore bounded, the Arzel`a-Ascoli theorem then gives a subsequence of theseorbits converging to a reparametrization of γ , contradicting the assumption that γ is isolated.Arguing by contradiction, suppose now that there exists a sequence s ′ k → ∞ with u M ( s k , · )
6∈ U for all k . We can nonetheless restrict to a subsequence for which u M ( s ′ k , · ) converges to some Reeb orbit ˜ γ : R /T Z → M ± . Then ˜ γ is disjoint from γ , and by continuity, one can find a sequence s ′′ k → ∞ for which each u M ( s ′′ k , V some fixed distance away from im γ . There must then be asubsequence for which u M ( s ′′ k , · ) converges to another T -periodic orbit, but this isimpossible since no such orbits exist in V \ im γ . (cid:3) To motivate the SFT compactness theorem, we shall now discuss three examplesof phenomena that can prevent a sequence of holomorphic curves from having acompact subsequence. The theorem will then tell us that these three things are, inessence, the only things that can go wrong.Throughout this section and the next, assume J k → J ∈ J τ ( ω h , r , H + , H − )is a C ∞ -convergent sequence of tame almost complex structures on the completedcobordism c W . More generally, one can also allow the data ω , h and H ± to varyin C ∞ -convergent sequences, but let’s not clutter the notation too much. We shalldenote the restrictions of J to the cylindrical ends by J + := J | [ r , ∞ ) × M + ∈ J ( H + ) , J − := J | ( −∞ , − r ] × M − ∈ J ( H − ) . Suppose u k := [(Σ k , j k , Γ + k , Γ − k , Θ k , u k )] ∈ M g,m ( J k , A k , γ + , γ − )is a sequence of J k -holomorphic curves in c W with fixed genus g ≥ m ≥ A k ∈ H ( W, ¯ γ + ∪ ¯ γ − ) and fixedcollections of asymptotic orbits γ ± = ( γ ± , . . . , γ ± m ± ). Observe that the energies E ( u k ) depend only on the orbits γ ± and relative homology classes A k , so in partic-ular, E ( u k ) is uniformly bounded whenever the relative homology class is also fixed.The fundamental question of this section is: ectures on Symplectic Field Theory Question. If E ( u k ) is uniformly bounded and no subsequence of u k converges toan element of M g,m ( J, A, γ + , γ − ) for any A ∈ H ( W, ¯ γ + ∪ ¯ γ − ) , what can happen? Suppose (Σ k , j k , Γ + k , Γ − k , Θ k ) = (Σ , j, Γ + , Γ − , θ ) is a fixed se-quence of domains, and choose Riemannian metrics on ˙Σ = Σ \ Γ and c W that aretranslation-invariant on the cylindrical ends of both. Suppose there exists a point ζ ∈ ˙Σ such that u k ( ζ ) is contained in a compact subset for all k . Suppose alsothat the maps u k : ˙Σ → c W are locally C -bounded outside some finite subsetΓ ′ = { ζ , . . . , ζ N } ⊂ ˙Σ , i.e. for every compact set K ⊂ ˙Σ \ Γ ′ , there exists a constant C K > k such that | du k | ≤ C K on K. Then elliptic regularity gives a subsequence that converges in C ∞ loc ( ˙Σ \ Γ ′ ) to a J -holomorphic curve u ∞ : ˙Σ \ Γ ′ → c W with E ( u ∞ ) ≤ lim sup E ( u k ) < ∞ , thus all the punctures Γ + ∪ Γ − ∪ Γ ′ of u ∞ areeither removable or positively or negatively asymptotic to Reeb orbits. We cannotbe sure that the asymptotic behavior of u ∞ at Γ ± is the same as for u k , but let’sassume this for now ( § u ∞ is doing at the additional punctures Γ ′ , but also what is happening to u k near thesepoints as its first derivative blows up. For this we can apply the familiar rescalingtrick: choose for each ζ i a sequence z ik → ζ i such that | du k ( z ik ) | =: R k → ∞ , alongwith a sequence ǫ k → ǫ k R k → ∞ , and using Lemma 9.4, assume without lossof generality that | du k ( z ) | ≤ R k for all z in the ǫ k -ball about z ik . For convenience,we can choose a holomorphic coordinate system identifying a neighborhood of ζ i with D ⊂ C and placing ζ i at the origin, so z ik → ζ i withthe Euclidean metric. Now setting v ik ( z ) = u ( z ik + z/R k ) for z ∈ D ǫ k R k gives a sequence of J k -holomorphic maps v ik : D ǫ k R k → c W whose energies and firstderivatives are both uniformly bounded. As in the arguments of §
2, we now havethree possibilities: • If u ik ( z ik ) has a bounded subsequence, then the corresponding subsequenceof v ik converges in C ∞ loc ( C ) to a J -holomorphic plane v i ∞ : C → c W with finiteenergy. • If u k ( z ik ) has a subsequence diverging to {±∞} × M ± , then translating v ik by the R -action produces a limiting finite-energy plane v i ∞ in the posi-tive/negative symplectization R × M ± .Viewing C as the punctured sphere S \ {∞} , the singularity of v i ∞ at ∞ may beremovable, in which case v i ∞ extends to a J -holomorphic sphere and we say that u k has “bubbled off a sphere” at ζ i . Alternatively, v i ∞ may be positively or negativelyasymptotic to a Reeb orbit at ∞ . Chris Wendl
Figure 9.2 shows two scenarios that could occur for a sequence in which | du k | blows up at three points Γ ′ = { ζ , ζ , ζ } . Both scenarios show u ∞ with ζ and ζ as removable singularities and ζ as a negative puncture, but the behavior of thevarious v i ∞ reveals a wide spectrum of possibilities. In the lower-left picture, thepoints u k ( z k ) are bounded and bubble off a sphere v ∞ : S → c W . The picture showsthat v ∞ passes through u ∞ ( ζ ) at some point; this does not follow from our argumentso far, but in this situation one can use a more careful analysis of u k near ζ to showthat it must be true, i.e. “bubbles connect”. At ζ , we have u k ( z k ) → {−∞} × M − and v ∞ is a plane in R × M − with a positive puncture asymptotic to the same orbitas ζ ; the coincidence of these orbits is another detail that does not follow from theanalysis above but turns out to be true in the general picture. The situation at ζ allows two different interpretations: v ∞ could be the plane with negative end in R × M + , meaning u k ( z k ) → { + ∞} × M + , and the picture then shows an additionalplane in c W with a positive end approaching the same asymptotic orbit as v ∞ as wellas a point passing through u ∞ ( ζ ). One would need to choose a different rescaledsequence near ζ to find this extra plane, but as we will see, the SFT compactnesstheorem dictates that some such object must be there. Alternatively, u k ( z k ) couldalso be bounded at ζ , in which case v ∞ must be the plane in c W with positiveend, and the extra plane above this is something that one could find via a differentchoice of rescaled sequence. In general, the range of actual possibilities can involvearbitrarily many additional curves that could be discovered via different choices ofrescaled sequences: e.g. there could be entire “bubble trees” as shown in the lower-right picture, where each v i ∞ is only one of several curves that arise as limits ofdifferent parametrizations of u k near ζ i . One good place to read about the analysisof bubble trees is [ HWZ03 , § Figure 9.2 already shows some phenomena that could be in-terpreted as “breaking” in the Floer-theoretic sense, but breaking can also happenwhen no derivatives are blowing up, simply due to the fact that our domains are non-compact. Figures 9.3 and 9.4 show three such scenarios, where we assume again that(Σ k , j k , Γ + k , Γ − k , Θ k ) = (Σ , j, Γ + , Γ − , Θ) is a fixed sequence of domains, and ˙Σ = Σ \ Γand c W carry Riemannian metrics that are translation-invariant on the cylindricalends such that | du k | ≤ C everywhere on ˙Σfor some constant C > k . This is a stronger condition than we hadin § ζ ∈ ˙Σ such that u k ( ζ ) is bounded, it impliesthat u ∞ converges in C ∞ loc ( ˙Σ) to a J -holomorphic map u ∞ : ˙Σ → c W with E ( u ∞ ) ≤ lim sup E ( u k ) < ∞ . Convergence in C ∞ loc is, however, not very strong:there may in general be no relation between the asymptotic behavior of u ∞ and u k at corresponding punctures, e.g. the top scenario in Figure 9.3 shows a case inwhich a negative puncture of u k becomes a removable singularity of u ∞ . Wheneverthis happens, there must be more to the story: in this example, one can choose ectures on Symplectic Field Theory c W c W c W R × M + R × M − R × M − R × M − ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ u k u k u ∞ u ∞ v ∞ v ∞ Figure 9.2.
Two possible pictures of spheres and/or planes thatcan bubble off when the first derivative blows up near three points.holomorphic cylindrical coordinates ( s, t ) ∈ ( −∞ , × S ⊂ ˙Σ near the negativepuncture of u k and find a sequence s k → ∞ such that the sequence of half-cylinders( −∞ , s k ] × S → c W : ( s, t ) u k ( s − s k , t ) Chris Wendl is uniformly C -bounded and thus converges in C ∞ loc ( R × S ) to a finite-energy J -holomorphic cylinder v − : R × S → c W . In the picture, v − turns out to have aremovable singularity at + ∞ mapping to the same point as the removable singularityof u ∞ , and its negative puncture approaches the same orbit as the negative punctureof u k .More complicated things can happen in general: the bottom scenario in this samefigure shows a case where all three singularities of u ∞ are removable, thus it extendsto a closed curve, while at one of the positive cylindrical ends [0 , ∞ ) × S ⊂ ˙Σ of u k , we can find a sequence s k → ∞ such that the half-cylinders[ − s k , ∞ ) × S → c W : ( s, t ) u k ( s + s k , t )are uniformly C -bounded and converge in C ∞ loc ( R × S ) to a J -holomorphic cylinder v : R × S → c W with one removable singularity and one positive puncture. Atthe other positive end, we can perform the same trick in two distinct ways for twosequences s k → ∞ , one diverging faster than the other: the result is a pair of J -holomorphic cylinders v , v : R × S → c W , the former with both singularitiesremovable (thus forming a holomorphic sphere in the picture), and the latter withone removable singularity and one positive puncture.It can get weirder. Remember that c W is also noncompact!In each of the above scenarios, we tacitly assumed that all of the various se-quences obtained by reparametrizing portions of u k were locally C -bounded, thusall of the limits were curves in c W . But it may also happen that some of these se-quences are C -bounded while others locally diverge toward {±∞} × M ± ; in fact,two such sequences that both diverge toward, say, { + ∞} × M + , might even locallydiverge infinitely far from each other , meaning one of them approaches { + ∞} × M + quantitatively faster than the other. This phenomenon leads to the notion of limitingcurves with multiple levels .In Figure 9.4, we see a scenario in which u k satisfies the same conditions as above,except that instead of u k ( ζ ) being bounded, it diverges to { + ∞} × M + . It followsthat after applying suitable R -translations, a subsequence converges in C ∞ loc ( ˙Σ) to a J + -holomorphic curve u ∞ : ˙Σ → R × M + with finite energy. In the example, all three of its punctures are nonremovable, buttwo of them approach orbits that have nothing to do with the asymptotic orbitsof u k . Now observe that since u k has a negative cylindrical end ( −∞ , × S ⊂ ˙Σ,one can necessarily find a sequence s k → ∞ such that u k ( − s k ,
0) is bounded, andthe sequence of half-cylinders( −∞ , s k ] × S → c W : ( s, t ) u k ( s − s k , t )is then uniformly C -bounded and thus has a subsequence convergent in C ∞ loc ( R × S )to a finite-energy J -holomorphic cylinder v : R × S → c W . In the picture, v hasboth a positive and a negative puncture, but its negative end again approaches adifferent Reeb orbit from the negative ends of u k , so one can deduce that theremust be still more happening near −∞ : there exists another sequence s ′ k → ∞ with ectures on Symplectic Field Theory u k u k u k u ∞ u ∞ v v v v − v − Figure 9.3.
Even with fixed conformal structures on the domainsand without bubbling, a sequence of punctured holomorphic curves in c W can break to produce multiple curves in c W with extra removablepunctures. The picture shows two such scenarios. s ′ k − s k → ∞ such that suitable R -translations of the half-cylinders( −∞ , s k ] × S → ( −∞ , − r ] × M − : ( s, t ) u k ( s − s ′ k , t )define uniformly C -bounded maps into R × M − , giving a subsequence that convergesin C ∞ loc ( R × S ) to a finite-energy J − -holomorphic cylinder v − : R × S → R × M − . Finally, the fact that u ∞ has a positive asymptotic orbit different from those of u k indicates that something more must also be happening near + ∞ : in the example,one of the positive ends [0 , ∞ ) × S ⊂ ˙Σ admits a sequence s k → ∞ such that u k ( s k , ∈ { r k } × M + for some r k → ∞ , and suitable R -translations of[ − s k , ∞ ) × S → [ r , ∞ ) × M + : ( s, t ) u k ( s + s k , t )become a uniformly C -bounded sequence of half-cylinders in R × M + , with a sub-sequence converging in C ∞ loc ( R × S ) to a finite-energy J + -holomorphic cylinder v : R × S → R × M + that connects the errant asymptotic orbit of u ∞ to the corresponding orbit of u k . Onecan now perform the same trick at the other positive end of ˙Σ, as there necessarilyalso exists a sequence s ′ k → ∞ in this end such that u k ( s ′ k , ∈ { r k } × M + forthe same sequence r k → ∞ as in the above discussion. The resulting limit curve Chris Wendl c W c W c W R × M + R × M + R × M − u k u k u ∞ v v v v − Figure 9.4.
Different portions of a breaking sequence of curves mayalso become infinitely far apart in the limit, so that some live in c W while others live in the symplectization of M + or M − . v : R × S → R × M + however is not guaranteed to be interesting: in the picture,it turns out to be a trivial cylinder.The type of degeneration shown in Figure 9.4 happens whenever the sequence u k does interesting things in multiple regions of its domain that are sent increasinglyfar away from each other in the image. The usual picture of c W that collapsesthe cylindrical ends to a finite size therefore becomes increasingly inadequate forvisualizing u k as k → ∞ : the middle picture in Figure 9.4 deals with this byexpanding the scale of the cylindrical ends so that the convergence to upper andlower levels becomes visible. We next needto relax the assumption that the Riemann surfaces (Σ k , j k , Γ + k ⊔ Γ − k ⊔ Θ k ) are fixed.Recall that for integers g ≥ ℓ ≥
0, the moduli space of pointed Riemannsurfaces is the space of equivalence classes M g,ℓ = { (Σ , j, Θ) } (cid:14) ∼ , where (Σ , j ) is a closed connected Riemann surface of genus g , Θ ⊂ Σ is an or-dered set of ℓ distinct points, and (Σ , j, Θ) ∼ (Σ ′ , j ′ , Θ ′ ) whenever there exists abiholomorphic map ϕ : (Σ , j ) → (Σ ′ , j ′ ) taking Θ to Θ ′ with the ordering preserved.This space is fairly easy to understand in the finitely many cases with 2 g + ℓ < M ,ℓ is a one-point space for each ℓ ≤
3. We say that (Σ , j,
Θ) is stable when-ever χ (Σ \ Θ) <
0, which means 2 g + ℓ ≥
3. In the stable case, one can showthat every pointed Riemann surface has a finite automorphism group, and M g,ℓ is a ectures on Symplectic Field Theory smooth orbifold of dimension 6 g − ℓ . It is generally not compact, but it admitsa natural compactification M g,ℓ ⊃ M g,ℓ , known as the Deligne-Mumford compactification . We shall now give a sketchof this construction from the perspective of hyperbolic geometry; for more details,see [
Hum97 , SS92 ].We recall first the following standard result.
Theorem (Uniformization theorem) . Every simply connected Riemann surfaceis biholomorphically equivalent to either the Riemann sphere S = C ∪ {∞} , thecomplex plane C or the upper half plane H = { Im z > } ⊂ C . The uniformization theorem implies that every Riemann surface can be presentedas a quotient of either ( S , i ), ( C , i ) or ( H , i ) by some freely acting discrete groupof biholomorphic transformations. The only punctured surface ˙Σ = Σ \ Θ that has S as its universal cover is S itself. It is almost as easy to see which surfaces arecovered by C , as the only biholomorphic transformations on ( C , i ) with no fixedpoints are the translations, so every freely acting discrete subgroup of Aut( C , i ) iseither trivial, a cyclic group of translations or a lattice. The resulting quotients are,respectively, ( C , i ), ( R × S , i ) ∼ = ( C \ { } , i ) and the unpunctured tori ( T , j ). All stable pointed Riemann surfaces are thus quotients of ( H , i ). Proposition . There exists on ( H , i ) a complete Riemannian metric g P ofconstant curvature − that defines the same conformal structure as i and has theproperty that all conformal transformations on ( H , i ) are also isometries of ( H , g P ) . Proof.
We define g P at z = x + iy ∈ H by g P = 1 y g E , where g E is the Euclidean metric. The conformal transformations on ( H , i ) are givenby fractional linear transformationsAut( H , i ) = (cid:26) ϕ ( z ) = az + bcz + d (cid:12)(cid:12)(cid:12) a, b, c, d ∈ R , ad − bc = 1 (cid:27) (cid:30) {± } = SL(2 , R ) / {± } =: PSL(2 , R ) , and one can check that each of these defines an isometry with respect to g P . Onecan also compute that g P has curvature −
1, and the geodesics of g P are preciselythe lines and semicircles that meet R orthogonally, parametrized so that they existfor all forward and backward time, thus g P is complete. For more details on all ofthis, the book by Hummel [ Hum97 ] is highly recommended. (cid:3)
By lifting to universal covers, this implies the following.
Corollary . For every pointed Riemann surface (Σ , j, Θ) such that χ (Σ \ Θ) < , the punctured Riemann surface (Σ \ Θ , j ) admits a complete Riemannianmetric g j of constant curvature − that defines the same conformal structure as j ,and has the property that all biholomorphic transformations on (Σ \ Θ , j ) are alsoisometries of (Σ \ Θ , g j ) . Chris Wendl ====
Figure 9.5.
Two distinct pair-of-pants decompositions for the samegenus 1 Riemann surface with three marked points. The decomposi-tions are shown from two perspectives: the pictures at the right aremeant to give a more accurate impression of the Poincar´e metric,which becomes singular and forms a cusp at each marked point.The metric g j in this corollary is often called the Poincar´e metric . It is uniquelydetermined by j .Every class in π ( ˙Σ) contains a unique geodesic for g j . Now suppose C ⊂ ˙Σ is aunion of disjoint embedded geodesics such that each connected component of ˙Σ \ C has the homotopy type of a disk with two holes. The components are then called singular pairs of pants , and the result is called a pair-of-pants decomposition of ( ˙Σ , j ). Two examples for the case g = 1 and ℓ = 3 are shown in Figure 9.5.A pair-of-pants decomposition for (Σ , j, Θ) gives rise to a local parametrizationof M g,ℓ near [(Σ , j, Θ)], known as the
Fenchel-Nielsen coordinates . These consistof two parameters that can be associated to each of the geodesics γ ⊂ Σ in thedecomposition, namely the length ℓ ( γ ) > twist parameter θ ( γ ) ∈ S , which describes how the two neighboring pairs of pants are glued togetheralong γ . Note that by computing Euler characteristics, there are always exactly − χ (Σ \ Θ) = 2 g − ℓ pairs of pants in a decomposition, so that the total numberof geodesics involved is [3(2 g − ℓ ) − ℓ ] / g − ℓ , thus one can read off theformula dim M g,ℓ = 6 g − ℓ from this geometric picture.One can also see the noncompactness of M g,ℓ in this picture quite concretely:the twist parameters belong to a compact space, but each length parameter can ectures on Symplectic Field Theory potentially shrink to 0 or blow up to ∞ as j (and hence g j ) is deformed. It turnsout that the latter possibility is an illusion, but one may need to switch to a differentpair-of-pants decomposition to see why: Theorem.
For every pair of integers g ≥ and ℓ ≥ with g + ℓ ≥ , thereexists a constant C = C ( g, ℓ ) > such that every [(Σ , j, Θ)] ∈ M g,ℓ admits a pair-of-pants decomposition in which all geodesics bounding the pairs of pants have lengthat most C . This theorem implies that from a hyperbolic perspective, the only meaningfulway for stable pointed Riemann surfaces to degenerate is when some of the boundinggeodesics in a pair-of-pants decomposition shrink to length zero. Figure 9.6 showsseveral examples of degenerate Riemann surfaces that can arise in this way for g = 1and ℓ = 3, giving elements of the space that we will now define as M , . Definition . A nodal Riemann surface with ℓ ≥ N ≥ nodes is a tuple ( S, j, Θ , ∆) consisting of: • A closed but not necessarily connected Riemann surface (
S, j ); • An ordered set of ℓ points Θ ⊂ S ; • An unordered set of 2 N points ∆ ⊂ S \ Θ equipped with an involution σ : ∆ → ∆. Each pair { z, σ ( z ) } for z ∈ ∆ is referred to as a node .Let b S denote the closed surface obtained by performing connected sums on S ateach node { z + , z − } ⊂ ∆. We then say that ( S, j, Θ , ∆) is connected if and only if b S is connected, and the genus of b S is called the arithmetic genus of ( S, j, Θ , ∆).We say that ( S, j, Θ , ∆) is stable if every connected component of S \ (Θ ∪ ∆)has negative Euler characteristic. Finally, two nodal Riemann surfaces ( S, j, Θ , ∆)and ( S ′ , j ′ , Θ ′ , ∆ ′ ) are considered equivalent if there exists a biholomorphic map ϕ : ( S, j ) → ( S ′ , j ′ ) taking Θ to Θ ′ with the ordering preserved and taking ∆ to ∆ ′ such that nodes are mapped to nodes.The nodes { z + , z − } ⊂ ∆ are typically represented in pictures as self-intersectionsof S , cf. Figure 9.6. We can think of the stable nodal surfaces as precisely thosewhich admit (possibly singular) pair-of-pants decompositions. All nodal Riemannsurfaces we consider will be assumed connected in the sense defined above unlessotherwise noted; note that S itself can nonetheless be disconnected, as is the casein four out of the six nodal surfaces shown in Figure 9.6.We now introduce some further terminology and notation that will be usefulin the next section as well. Whenever ˙Σ = Σ \ Γ is obtained by puncturing aRiemann surface (Σ , j ) at finitely many points Γ ⊂ Σ, we shall define the circlecompactification
Σ := ˙Σ ∪ [ z ∈ Γ δ z , where for each z ∈ Γ, the circle δ z is defined as a “half-projectivization” of thetangent space at z : δ z := ( T z Σ \ { } ) . R ∗ + , Chris Wendl
Figure 9.6.
Starting from each of the pair-of-pants decompositionsfor the g = 1 and ℓ = 3 case from Figure 9.5, shrinking geodesiclengths to zero produces various examples of stable nodal Riemannsurfaces belonging to M , .with the positive real numbers R ∗ + acting by scalar multiplication. To understandthe topology of Σ, one can equivalently define it by choosing holomorphic cylindricalcoordinates [0 , ∞ ) × S ⊂ ˙Σ near each z , and replacing the open half-cylinder with[0 , ∞ ] × S , where δ z is now the circle at infinity {∞} × S . There is no naturalchoice of global smooth structure on Σ, but it is homeomorphic to an oriented surfacewith boundary and carries both smooth and conformal structures on its interior, dueto the obvious identification ˙Σ = Σ \ [ z ∈ Γ δ z ⊂ Σ . ectures on Symplectic Field Theory The conformal structure of Σ at each z ∈ Γ does induce on each of the circles δ z an orthogonal structure , meaning a preferred class of homeomorphisms to S thatare all related to each other by rotations. One can therefore speak of orthogonalmaps δ z → δ z ′ for z, z ′ ∈ Γ, which are always homeomorphisms and can eitherpreserve or reverse orientation.Now if (
S, j, Θ , ∆) is a nodal Riemann surface, we let ˙ S = S \ ∆ and form thecircle compactification S , which has the topology of a compact oriented surface withboundary. Given a node { z + , z − } ⊂ ∆, a decoration for { z + , z − } is a choice oforientation reversing orthogonal mapΦ : δ z + → δ z − . We say that (
S, j, Θ , ∆) is a decorated nodal surface if it is equipped with achoice of decoration Φ for every node, or partially decorated if Φ is defined forsome subset of the nodes. A partial decoration Φ gives rise to another compactoriented surface b S Φ := S (cid:14) ∼ , where the equivalence relation identifies δ z + with δ z − via Φ for each decorated node { z + , z − } ⊂ ∆. Note that if every node is decorated, then b S Φ has the topology of aclosed connected and oriented surface whose genus defines the arithmetic genus of( S, j, Θ , ∆) according to Definition 9.21. We shall denote the collection of specialcircles in b S Φ where boundray components δ z + , δ z − ⊂ ∂S have been identified by C Φ ⊂ b S Φ . Since b S Φ \ ( ∂ b S Φ ∪ C Φ ) has a natural identification with ˙ S , it inherits smooth andconformal structures which degenerate along C Φ and ∂ b S Φ . We will say that twopartially decorated nodal Riemann surfaces ( S, j, Θ , ∆ , Φ) and ( S ′ , j ′ , Θ ′ , ∆ ′ , Φ ′ ) are equivalent if ( S, j, Θ , ∆) and ( S ′ , j ′ , Θ ′ , ∆ ′ ) are equivalent via a biholomorphic map ϕ : ( S, j ) → ( S ′ , j ′ ) that extends continuously from ˙ S → ˙ S ′ to a homeomorphism b S Φ → b S ′ Φ ′ .Now if 2 g + ℓ ≥
3, define M g,ℓ as the set of equivalence classes of stable nodalRiemann surfaces with ℓ marked points and arithmetic genus g . There is a naturalinclusion M g,ℓ ⊂ M g,ℓ by regarding each pointed Riemann surface (Σ , j, Θ) as a nodal Riemann surface(Σ , j, Θ , ∆) with ∆ = ∅ . The most important property of M g,ℓ is that it admitsthe structure of a compact metrizable topological space for which the inclusion M g,ℓ ֒ → M g,ℓ is continuous onto an open subset. Rather than formulating all ofthis in precise terms, let us state the main corollary that is important to know inpractice. Theorem . Fix g ≥ and ℓ ≥ with g + ℓ ≥ . Then for any sequence [(Σ k , j k , Θ k )] ∈ M g,ℓ , there exists a stable nodal Riemann surface [( S, j, Θ , ∆)] ∈M g,ℓ such that after restricting to a subsequence, [(Σ k , j k , Θ k )] → [( S, j, Θ , ∆)] Chris Wendl in the following sense: ( S, j, Θ , ∆) admits a decoration Φ such that for sufficientlylarge k , there are homeomorphisms ϕ : b S Φ → Σ k , smooth outside of C Φ , which map Θ to Θ k preserving the ordering and satisfy ϕ ∗ j k → j in C ∞ loc ( b S Φ \ C ∆ ) . As one might gather from the above statement, one could just as well define acompact metrizable topology on the space of equivalence classes of decorated nodalRiemann surfaces and then characterize the topology of M g,ℓ via the natural pro-jection that forgets the decorations. Exercise . The space M , has a natural identification with S \ { , , ∞} ,defined by choosing the unique identification of any 4-pointed Riemann sphere( S , j, ( z , z , z , z )) with C ∪ {∞} such that z , z , z are identified with 0 , , ∞ respectively, while z is sent to some point in S \ { , , ∞} . Show that this extendscontinuously to an identification of M , with S . What do the three nodal curvesin M , \ M , look like in terms of pair-of-pants decompositions? We now introduce the natural compactification of M g,m ( J, A, γ + , γ − ). A punctured J -holomorphic nodal curve in ( c W , J )with m ≥ S, j, Γ + , Γ − , Θ , ∆ , u ), where • ( S, j, Γ + ⊔ Γ − ⊔ Θ , ∆) is a nodal Riemann surface, with | Θ | = m ; • u : ( ˙ S, j ) → ( c W , J ) for ˙ S := S \ (Γ + ∪ Γ − ) is an asymptotically cylindrical J -holomorphic map with positive punctures Γ + and negative punctures Γ − such that for each node { z + , z − } ⊂ ∆, u ( z + ) = u ( z − ).Equivalence of two nodal curves( S , j , Γ +0 , Γ − , Θ , ∆ , u ) ∼ ( S , j , Γ +1 , Γ − , Θ , ∆ , u )is defined as the existence of an equivalence of nodal Riemann surfaces ϕ : ( S , j , Γ +0 ⊔ Γ − ⊔ Θ , ∆ ) → ( S , j , Γ +1 ⊔ Γ − ⊔ Θ , ∆ ) such that u = u ◦ ϕ . We say that( S, j, Γ + , Γ − , Θ , ∆ , u ) is connected if and only if the nodal Riemann surface ( S, j, Γ + ⊔ Γ − ⊔ Θ , ∆) is connected, and its arithmetic genus is then defined to be the arith-metic genus of the latter. We say that ( S, j, Γ + , Γ − , Θ , ∆ , u ) is stable if every con-nected component of S \ (Γ + ∪ Γ − ∪ Θ ∪ ∆) on which u is constant has negative Eulercharacteristic. Note that the underlying nodal Riemann surface ( S, j, Γ + ⊔ Γ − ⊔ Θ , ∆)need not be stable in general.Nodal curves are sometimes also referred to as holomorphic buildings of height 1 .These are the objects that form the Gromov compactification of M g,m ( J, A ) when W is a closed symplectic manifold. One can now roughly imagine how the compactnesstheorem in that setting is proved: given a converging sequence of almost complexstructures J k → J and a sequence [(Σ k , j k , Θ k , u k )] ∈ M g,m ( J k , A k ) with uniformlybounded energy, we can first add some auxiliary marked points if necessary to assumethat 2 g + m ≥
3. Now a subsequence of the domains [(Σ k , j k , Θ k )] ∈ M g,m converges ectures on Symplectic Field Theory to an element of the Deligne-Mumford space [( S, j, Θ , ∆)] ∈ M g,m . Concretely,this means that for large k , our sequence in M g,m ( J k , A k ) admits representatives(Σ , j ′ k , Θ , u ′ k ), with Σ a fixed surface with fixed marked points Θ ⊂ Σ, and (
S, j, Θ , ∆)admits decorations Φ so that one can identify b S Φ with Σ and find j ′ k → j in C ∞ loc (Σ \ C )for some collection of disjoint circles C ⊂ Σ. The connected components of (Σ \ C, j )are then biholomorphically equivalent to the connected components of ( S \ ∆ , j ),and if the newly reparametrized maps u ′ k : Σ → W are uniformly C -bounded onΣ \ C , then a subsequence converges in C ∞ loc (Σ \ C ) to a limiting finite-energy J -holomorphic map u ∞ : ( S \ ∆ , j ) → ( W, J ), whose singularities at ∆ are removable.In particularly nice cases, this may be the end of the story, and our subsequenceof [(Σ k , j k , Θ k , u k )] ∈ M g,m ( J k , A k ) converges to the nodal curve [( S, j, Θ , ∆ , u ∞ )];in particular the domain [( S, j, Θ , ∆)] in this case is stable and is thus an elementof M g,m . But more complicated things can also happen, e.g. u ′ k might not be C -bounded, in which case there is bubbling. The bubbles that arise will be eitherplanes or spheres, so they produce extra domain components with nonnegative Eulercharacteristic, but since they are never constant, the limiting nodal curve is stillconsidered stable. Similarly, since Σ \ C is not compact, there can also be breaking asin Figure 9.3, producing more non-stable domain components which can be cylindersin addition to planes and spheres—but again, the limiting map on these componentswill never be constant. Only a small subset of the phenomena ob-served in § g, m, N + , N − ≥
0, a holomorphic building of height N − | | N + with arithmetic genus g and m marked points is a tuple u = ( S, j, Γ + , Γ − , Θ , ∆ nd , ∆ br , L, Φ , u ) , with the various data defined as follows: • The domain ( S, j, Γ + ⊔ Γ − ⊔ Θ , ∆ nd ⊔ ∆ br ) is a connected but not necessarilystable nodal Riemann surface of arithmetic genus g , where | Θ | = m , andthe involution on ∆ nd ⊔ ∆ br is assumed to preserve the subsets ∆ nd and ∆ br .Matched pairs in these subsets are called the nodes and breaking pairs respectively of u . The marked points of u are the points in Θ, while Γ + and Γ − are its positive and negative punctures respectively. • The level structure is a locally constant function L : S → {− N − , . . . , − , , , . . . , N + } that attains every value in {− N − , . . . , N + } except possibly 0, and satisfies:(1) L ( z + ) = L ( z − ) for each node { z + , z − } ⊂ ∆ nd ;(2) Each breaking pair { z + , z − } ⊂ ∆ br can be labelled such that L ( z + ) − L ( z − ) = 1;(3) L (Γ + ) = { N + } and L (Γ − ) = {− N − } . Chris Wendl • The decoration is a choice of orientation-reversing orthogonal map δ z + Φ −→ δ z − for each breaking pair { z + , z − } ⊂ ∆ br . • The map is an asymptotically cylindrical pseudoholomorphic curve u : ( ˙ S := S \ (Γ + ∪ Γ − ∪ ∆ br ) , j ) → G N ∈{− N − ,...,N + } ( c W N , J N ) , where( c W N , J N ) := ( R × M + , J + ) for N ∈ { , . . . , N + } , ( c W , J ) for N = 0 , ( R × M − , J − ) for N ∈ {− N − , . . . , − } , and u sends ˙ S ∩ L − ( N ) into c W N for each N , with positive punctures atΓ + and negative punctures at Γ − . Moreover, u ( z + ) = u ( z − ) for every node { z + , z − } ⊂ ∆ nd , and for each breaking pair { z + , z − } ⊂ ∆ br labelled with L ( z + ) − L ( z − ) = 1, u has a positive puncture at z − and a negative puncture at z + asymptoticto the same orbit, such that if u + : δ z + → M ± and u − : δ z − → M ± denotethe induced asymptotic parametrizations of the orbit, then u + = u − ◦ Φ : δ z + → M ± . The following additional notation and terminology for the building u will beuseful to keep in mind. For each N ∈ {− N − , . . . , , . . . , N + } , denote˙ S N := (cid:0) S \ (Γ + ∪ Γ − ∪ ∆ br ) (cid:1) ∩ L − ( N ) , and denote the restriction of u to this subset by u N : ˙ S N → R × M + if N > , c W if N = 0 , R × M − if N < . Including Θ ∩ L − ( N ) and ∆ nd ∩ L − ( N ) in the data defines u N as a (generally dis-connected) nodal curve with marked points, whose positive punctures are in bijectivecorrespondence with the negative punctures of u N +1 if N < N + . We call u N the N th level of u , and all it an upper or lower level if N >
N < main level if N = 0. By convention, every holomorphic building in c W hasexactly one main level (which lives in c W itself) and arbitrary nonnegative numbersof upper and lower levels (which live in the symplectizations R × M ± ). One slightlysubtle detail is that it is possible for the main level to be empty , meaning 0 is not inthe image of the level function L . The requirement that L should attain every othervalue from − L − to L + is a convention to ensure that upper and lower levels are notempty, so e.g. if a building has an empty main level and no lower levels, then thelowest nonempty upper level is always labelled 1 instead of something arbitrary. ectures on Symplectic Field Theory The positive punctures of the topmost level of u are Γ + , and the negative punc-tures of the bottommost level are Γ − , so these give rise to lists of positive/negativeasymptotic orbits γ ± = ( γ ± , . . . , γ ± k ± ) in M ± . There is also a relative homology class[ u ] ∈ H ( W, ¯ γ + ∪ ¯ γ − ) . To define this, recall from § u : ˙Σ → c W : weconsidered the retraction π : c W → W that collapses each cylindrical end to M ± ⊂ ∂W , and noted that since u is asymptotically cylindrical, the map π ◦ u : ˙Σ → W extends to a continuous map on the circle compactification,¯ u : Σ → W, whose relative homology class gives the definition of [ u ]. The conditions on nodesand breaking orbits allow us to perform a similar trick for the building u , using themap π : G N ∈{− N − ,...,N + } c W N → W which acts as the identity on W but collapses cylindrical ends of c W to ∂W andsimilarly collapses each copy of R × M ± to M ± ⊂ ∂W . Extending the decora-tions Φ arbitrarily to decorations of the nodes ∆ nd , one can then take the circlecompactification of ˙ S := S \ (Γ + ∪ Γ − ∪ ∆ nd ∪ ∆ br ) and glue matching boundarycomponents together along Φ to form a compact surface with boundary S Φ suchthat π ◦ u : ˙ S → W extends to a continuous map¯ u : S Φ → W. Its relative homology class defines [ u ] ∈ H ( W, ¯ γ + ∪ ¯ γ − ).We say that the building u is stable if two properties hold:(1) Every connected component of S \ (Γ + ∪ Γ − ∪ Θ ∪ ∆ nd ∪ ∆ br ) on which themap u is constant has negative Euler characteristic;(2) There is no N ∈ {− N − , . . . , N + } for which the N th level consists entirelyof a disjoint union of trivial cylinders without any marked points or nodes.An equivalence between two holomorphic buildings u i = ( S i , j i , Γ + i , Γ − i , Θ i , ∆ nd i , ∆ br i , L i , Φ i , u i ) , i = 0 , S , j , Γ +0 ⊔ Γ +0 ⊔ Θ , ∆ nd0 ⊔ ∆ br0 , Φ ) ϕ −→ ( S , j , Γ +1 ⊔ Γ +1 ⊔ Θ , ∆ nd1 ⊔ ∆ br1 , Φ )such that ϕ (Γ ± ) = Γ ± , ϕ (Θ ) = Θ , ϕ (∆ nd0 ) = ∆ nd1 , ϕ (∆ br0 ) = ∆ br1 , L ◦ ϕ = L , and u ◦ ϕ = u , while u N ◦ ϕ = u N up to R -translation for each N = 0 . Given lists of orbits γ ± and a relative homology class A , the set of equivalenceclasses of stable holomorphic buildings in ( c W , J ) with arithmetic genus g and m Chris Wendl marked points, positively/negatively asymptotic to γ ± and homologous to A will bedenoted by M g,m ( J, A, γ + , γ − ) . Observe that for any A = 0, there is a natural inclusion M g,m ( J, A, γ + , γ − ) ⊂M g,m ( J, A, γ + , γ − ) defined by regarding J -holomorphic curves in M g,m ( J, A, γ + , γ − )as buildings with no upper or lower levels and no nodes. Such buildings are alwaysstable if A = 0 because they are not constant. For a general definition of the topology of M g,m ( J, A, γ + , γ − )and the proof that it is both compact and metrizable, we refer to [ BEH + ] or themore comprehensive treatment in [ Abb14 ]. The following statement contains allthe details about the topology that one usually needs to know in practice (see Fig-ure 9.7).
Theorem . Fix integers g ≥ and m ≥ , and assume all Reeb orbits in ( M, H + ) and ( M, H − ) are nondegenerate. Then for any sequence [(Σ k , j k , Γ + k , Γ − k , Θ k , u k )] ∈ M g,m ( J k , A k , γ + , γ − ) of nonconstant J k -holomorphic curves in c W with uniformly bounded energy E ( u k ) ,there exists a stable holomorphic building [ u ∞ ] = [( S, j, Γ + , Γ − , Θ , ∆ nd , ∆ br , L, Φ , u ∞ )] ∈ M g,m ( J, A, γ + , γ − ) such that after restricting to a subsequence, [(Σ k , j k , Γ + k , Γ − k , Θ k , u k )] → [ u ∞ ] in thefollowing sense. The decorations Φ at ∆ br can be extended to decorations at ∆ nd so that if b S Φ denotes the closed oriented topological -manifold obtained from S \ (∆ nd ∪ ∆ br ) by gluing circle compactifications along Φ , then for k sufficiently large,there exist homeomorphisms ϕ k : b S Φ → Σ k that are smooth outside of C Φ , map Γ + ⊔ Γ − ⊔ Θ to Γ + k ⊔ Γ − k ⊔ Θ k with the orderingpreserved, and satisfy ϕ ∗ k j k → j in C ∞ loc ( b S Φ \ C Φ ) . Moreover for N = {− N − , . . . , , . . . , N } , let v Nk := u k ◦ ϕ k | ¨ S N : ¨ S N → c W , with ¨ S N := (cid:0) S \ (Γ + ∪ Γ − ∪ ∆ nd ∪ ∆ br ) (cid:1) ∩ L − ( N ) regarded as a subset of b S Φ \ C Φ .Then:(1) v k → u N ∞ in C ∞ loc ( ¨ S N , c W ) ;(2) For each ± N > , v Nk has image in the positive/negative cylindrical end forall k sufficiently large, and there exists a sequence r Nk → ±∞ such that theresulting R -translations converge: τ − r Nk ◦ v Nk → u N ∞ in C ∞ loc ( ¨ S N , R × M ± ) . ectures on Symplectic Field Theory The rates of divergence of the sequences r Nk → ±∞ are related by r N +1 k − r Nk → + ∞ for all N < N + . Finally, let S Φ denote the compact topological surface with boundary defined as thecircle compactification of b S Φ \ (Γ + ∪ Γ − ) , and let Σ k denote the circle compactificationof ˙Σ k := Σ k \ (Γ + k ∪ Γ − k ) . Then for all k large, ϕ k extends to a continuous map ¯ ϕ k : S Φ → Σ k such that ¯ u k ◦ ¯ ϕ k → ¯ u ∞ in C ( S Φ , W ) . Remark . The theorem is also true under the more general hypothesisthat the Reeb vector fields are Morse-Bott. In this case, one can also allow theasymptotic Reeb orbits of the sequence to vary, as long as the sum of their periods isuniformly bounded—such a bound plays the role of an energy bound and guaranteesa convergent subsequence of orbits via the Arzel`a-Ascoli theorem.
Remark . Stability of the limit in Theorem 9.24 is guaranteed for the samereasons as in our discussion of Gromov compactness in § uniqueness of the limiting building for any convergent sequence, i.e. it isthe reason why M g,m ( J, A, γ + , γ − ) is a Hausdorff space. Indeed, if u k converges to astable building u ∞ , then under the notion of convergence described in the theorem,it will also converge to a building u ′∞ constructed out of u ∞ by adding to S an extraspherical component, attaching it to the rest by a single node and extending the map u ∞ to be constant on the extra component. One can also insert extra levels into u ∞ that consist only of trivial cylinders, and u k will still converge to the resultingbuilding. But these modifications produce buildings that are not stable and thusare not elements of M g,m ( J, A, γ + , γ − ). A few minor modifica-tions to the above discussion are necessary to compactify the moduli space of curvesin a symplectization ( R × M, J ) for J ∈ J ( H ). It is possible to view this as a specialcase of a completed symplectic cobordism, but this perspective produces a certainamount of extraneous data that is not meaningful. The key observation is that inthe presence of an R -action, one should really compactify M g,m ( J, A, γ + , γ − ) (cid:14) R instead of M g,m ( J, A, γ + , γ − ). The compactification M g,m ( J, A, γ + , γ − ) then con-sists of holomorphic buildings as defined in § R × M , there is no longer a distinguished main level or anymeaningful notion of upper vs. lower levels; the level structure is simply a function L : S → { , . . . , N } for some N ∈ N , and equivalence of buildings must permit R -translations within each level. For these reasons, the SFT compactness theoremin symplectizations has a few qualitative differences, but is still very much analogousto Theorem 9.24. Chris Wendl c W c W R × M + R × M − R × M − R × M − M + M − Figure 9.7.
Convergence to a building with arithmetic genus 2, oneupper level and three lower levels.To complete the picture, we should mention one more type of compactness theo-rem that appears in [
BEH + ], which is colloquially described as stretching theneck . The geometric idea is as follows: suppose ( W, ω ) is a closed symplecticmanifold and M ⊂ W is a stable hypersurface that separates W into two pieces W = W − ∪ M W + , with an induced stable Hamiltonian structure H = ( ω, λ ) thatorients M as the boundary of W − . A neighborhood of M in ( W, ω ) can then beidentified symplectically with( N ǫ , ω ǫ ) := (( − ǫ, ǫ ) × M, d ( rλ ) + ω )for sufficiently small ǫ >
0. The idea now is to replace N ǫ with larger collars of theform (( − T, T ) × M, d ( f ( r ) λ ) + ω ) , with C -small functions f chosen with f ′ > N ǫ , ω ǫ ). This collar looks like a piece of the symplectization of( M, H ), thus we are free to choose tame almost complex structures whose restrictionsto the inserted collar belong to J ( H ). Symplectic manifolds constructed in this The assumption that M ⊂ W separates W is inessential, but makes certain details in thisdiscussion more convenient. ectures on Symplectic Field Theory way are all symplectomorphic, but their almost complex structures degenerate asone takes T → ∞ . Given a sequence T k → ∞ and a corresponding degeneratingsequence J k , a sequence u k of J k -holomorphic curves with bounded energy convergesto yet another form of holomorphic building, this time involving a bottom level in c W − := W − ∪ M ([0 , ∞ ) × M ) with positive punctures approaching orbits in M , somefinite number of middle levels that live in the symplectization of M , and a top levelthat lives in c W + := (( −∞ , × M ) ∪ M W + with negative punctures approaching M .A very popular example for applications arises from Lagrangian submanifolds L ⊂ W . By the Weinstein neighborhood theorem, L always has a neighborhood W − symplectomorphic to a neighborhood of the zero-section in T ∗ L , so M := ∂W − is a contact-type hypersurface contactomorphic to the unit cotangent bundle of L .Stretching the neck then yields T ∗ L as the completion of W − , and W \ L as the com-pletion of W + := W \ ˚ W − . This construction has often been used in order to studyLagrangian submanifolds via SFT-type methods, see e.g. [ EGH00 , Theorem 1.7.5]and [
Eva10 , CM ].ECTURE 10 Cylindrical contact homology and the tight -toriContents T and Giroux torsion 19110.2. Definition of cylindrical contact homology 194 HC ∗ ( T , ξ k ) T , ξ k ) 219 We’ve now developed enough of the technical machinery of holomorphic curvesto be able to give a rigorous construction of the most basic version of SFT and applyit to a problem in contact topology. T and Giroux torsion As a motivating goal in this lecture, we will prove a result about the classificationof contact structures on T = S × S × S . Denote the three global coordinateson T valued in S = R / Z by ( ρ, φ, θ ), and for any k ∈ N , consider the contactstructure ξ k := ker α k , where α k := cos(2 πkρ ) dθ + sin(2 πkρ ) dφ. It is an easy exercise to verify that these all satisfy the contact condition α k ∧ dα k > Gir94 ] and Kanda [
Kan97 ]. Theorem . For each pair of positive integers k = ℓ , the contact manifolds ( T , ξ k ) and ( T , ξ ℓ ) are not contactomorphic. One of the reasons this result is interesting is that it cannot be proved usingany so-called “classical” invariants, i.e. invariants coming from algebraic topology.An example of a classical invariant would be the Euler class of the oriented vectorbundle ξ k → T , or anything else that depends only on the isomorphism class of this Chris Wendl ρ φθ
Figure 10.1.
The contact structures ξ k on T can be constructedby gluing k copies of the same model [0 , × T to each other cyclically.bundle. The following observation shows that such invariants will never distinguish ξ k from ξ ℓ . Proposition . For every k, ℓ ∈ N , ξ k and ξ ℓ are homotopic through asmooth family of oriented -plane fields on T . Proof.
In fact, all the ξ k can be deformed smoothly to ker dρ , via the homotopyker [(1 − s ) α k + s dρ ] , s ∈ [0 , . (cid:3) Remark . One can check in fact that the 1-form in the homotopy givenabove is contact for every s ∈ [0 , ξ k is isotopic to an arbitrarily small perturbation of the foliation ker dρ . In [ Gir94 ],Giroux used this observation to show that all of them are what we now call weaklysymplectically fillable . If ker dρ were also contact, then Gray’s theorem would implythat ξ k and ξ ℓ are always isotopic. Thus Theorem 10.1 indicates the impossibilityof modifying a homotopy from ξ k to ξ ℓ into one that passes only through contactstructures.Let us place this discussion in a larger context. Using the coordinates ( ρ, φ, θ )on R × T , a pair of smooth functions f, g : R → R gives rise to a contact form α = f ( ρ ) dθ + g ( ρ ) dφ whenever the function D ( ρ ) := f ( ρ ) g ′ ( ρ ) − f ′ ( ρ ) g ( ρ ) is everywhere positive. Indeed,we have α ∧ dα = D ( ρ ) dρ ∧ dφ ∧ dθ , and one easily derives a similar formula for theReeb vector field, R α = 1 D ( ρ ) [ g ′ ( ρ ) ∂ θ − f ′ ( ρ ) ∂ φ ] . The condition
D > f, g ) : R → R windscounterclockwise around the origin with its angular coordinate strictly increasing.The simplest special case is the contact form α GT := cos(2 πρ ) dθ + sin(2 πρ ) dφ, ectures on Symplectic Field Theory which matches the formula for α on T given above. Let ξ GT := ker α GT on R × T . Definition . The
Giroux torsion
GT(
M, ξ ) ∈ N ∪ { , ∞} of a contact3-manifold ( M, ξ ) is the supremum of the set of positive integers k such that thereexists a contact embedding (cid:0) [0 , k ] × T , ξ GT (cid:1) ֒ → ( M, ξ ) . We write GT(
M, ξ ) = 0 if no such embedding exists for any k , and GT( M, ξ ) = ∞ if it exists for all k . Example . The tori ( T , ξ k ) for k ≥ Z are contactomorphic to ( R × T , ξ GT ) /k Z ,with k Z acting by translation of the ρ -coordinate. Thus GT( T , ξ k ) ≥ k − T ⊂ ( M, ξ ) embedded in a contact 3-manifold is called pre-Lagrangian if a neighborhood of T in ( M, ξ ) admits a contactomorphism to a neighborhood of { } × T in ( R × T , ξ GT ), identifying T with { } × T . The neighborhood in R × T can be arbitrarily small, thus the existence of a pre-Lagrangian torus does not implyGT( M, ξ ) >
0; in fact, pre-Lagrangian tori always exist in abundance, e.g. as bound-aries of neighborhoods of transverse knots (using the contact model provided by thetransverse neighborhood theorem). But given any pre-Lagrangian torus T ⊂ ( M, ξ ),one can make a local modification of ξ near T to produce a new contact structure (upto isotopy) with positive Giroux torsion. Define ( M ′ , ξ ′ ) from ( M, ξ ) by replacingthe small neighborhood (( − ǫ, ǫ ) × T , ξ GT ) with (( − ǫ, ǫ ) × T , ξ GT ), then identify M ′ with M by a choice of compactly supported diffeomorphism ( − ǫ, ǫ ) → ( − ǫ, ǫ ).There is now an obvious contact embedding of ([0 , × T , ξ GT ) into ( M, ξ ′ ), henceGT( M, ξ ′ ) ≥
1. Moreover, one can adapt the proof of Prop. 10.2 above to showthat ξ ′ is homotopic to ξ through a smooth family of oriented 2-plane fields. Theoperation changing ξ to ξ ′ is known as a Lutz twist along T . In this language, wesee that for each k ∈ N , ( T , ξ k +1 ) is obtained from ( T , ξ k ) by performing a Lutztwist along { } × T .The invariant GT( M, ξ ) is easy to define, but hard to compute in general. Thenatural guess, GT( T , ξ k ) = k − , turns out to be correct, as was shown in [ Gir00 ], so this is one way to proveTheorem 10.1, but not the approach we will take. The following example showsthat one must in any case be careful with such guesses.
Example . For each k ∈ N , define a model of S × S by S × S ∼ = (cid:0) [0 , k + 1 / × T (cid:1) (cid:14) ∼ where the equivalence relation identifies ( ρ, φ, θ ) ∼ ( ρ, φ ′ , θ ) for ρ ∈ { , k + 1 / } andevery θ, φ, φ ′ ∈ S . Near ρ = 0 and ρ = k +1 /
2, this means thinking of ( ρ, φ ) as polarcoordinates, so the two subsets { ρ = 0 } and { ρ = k + 1 / } become circles of theform S × { const } embedded in S × S . Since the φ -coordinate is singular at thesetwo circles, the contact form α GT needs to be modified slightly in this region beforeit will descend to a smooth contact form on S × S : this can be done by a C -small Chris Wendl modification of the form f ( ρ ) dθ + g ( ρ ) dφ , and the resulting contact structure isthen uniquely determined up to isotopy. We shall call this contact manifold( S × S , ξ k ) . Now observe that for each k ∈ N , ( S × S , ξ k +1 ) is obtained from ( S × S , ξ k ) bya Lutz twist. However, both contact manifolds are also overtwisted : recall that acontact 3-manifold ( M, ξ ) is overtwisted whenever it contains an embedded closed2-disk
D ⊂ M such that T ( ∂ D ) ⊂ ξ but T D| ∂ D ⋔ ξ . (Exercise: find a disk with thisproperty in ( S × S , ξ k )!) Eliashberg’s flexibility theorem for overtwisted contactstructures [ Eli89 ] implies that whenever ξ and ξ ′ are two contact structures on aclosed 3-manifold that are both overtwisted and are homotopic as oriented 2-planefields, they are actually isotopic. As a consequence, the contact structures ξ k on S × S defined above for every k ∈ N are all isotopic to each other. As tends to bethe case with most interesting h-principles, the isotopy is very hard to see concretely,but it must exist. Exercise . Show that if (
M, ξ ) is a closed overtwisted contact 3-manifold,then GT(
M, ξ ) = ∞ .In contrast to the S × S example above, the contact manifolds ( T , ξ k ) are notovertwisted, they are tight —in fact, the classification of contact structures on T by Giroux [ Gir94 , Gir99 , Gir00 ] and Kanda [
Kan97 ] states that these are all ofthe tight contact structures on T up to contactomorphism. We will use cylindricalcontact homology to show that they are not contactomorphic to each other. Thereader should keep Example 10.6 in mind and try to spot the reason why the sameargument cannot work for ( S × S , ξ k ). Remark . It has been conjectured that the converse of Exercise 10.7 mightalso hold, so every closed tight contact 3-manifold would have finite Giroux torsion.This conjecture is wide open.
Cylindrical contact homology is the natural“first attempt” at using holomorphic curves in symplectizations to define a Floer-type invariant of contact manifolds (
M, ξ ). The idea is to define a chain complexgenerated by Reeb orbits in M and a differential ∂ that counts holomorphic cylindersin R × M . We already know some pretty good reasons why this idea cannot workin general: in order to prove ∂ = 0, we need to be able to identify the space ofrigid “broken” holomorphic cylinders (these are what is counted by ∂ ) with theboundary of the compactified 1-dimensional space of index 2 cylinders (up to R -translation). But this compactified boundary has more than just broken cylindersin it, see Figure 10.2. In order to define cylindrical contact homology, one musttherefore restrict to situations in which complicated pictures like Figure 10.2 cannotoccur. The first useful remark in this direction is that since we are working witha stable Hamiltonian structure of the form ( dα, α ) for a contact form α , a certainsubset of the scenarios allowed by the SFT compactness theorem can be excludedimmediately. Indeed: ectures on Symplectic Field Theory Figure 10.2.
A family of holomorphic cylinders can converge inthe SFT topology to buildings that include more complicated curvesthan cylinders—this is why cylindrical contact homology is not welldefined for all contact manifolds.
Proposition . If J ∈ J ( α ) and u : ( ˙Σ , j ) → ( R × M, J ) is an asymptoticallycylindrical J -holomorphic curve, then u has at least one positive puncture. Let us give two proofs of this result, since both contain useful ideas. As prepara-tion for the first proof, recall the definition of energy for curves in symplectizationsof contact manifolds that we wrote down in Lecture 1: E ( u ) := sup f ∈T Z ˙Σ u ∗ d ( e f ( r ) α ) , where T := (cid:8) f ∈ C ∞ ( R , ( − , (cid:12)(cid:12) f ′ > (cid:9) . This formula is not identical to the definition of energy used in Lecture 9, but itis equivalent in the sense that any uniform bounds on one imply similar uniformbounds on the other.
First proof of Proposition 10.9.
Denote the positive and negative punc-tures of u : ˙Σ → R × M by Γ + and Γ − respectively, and suppose u is asymptoticat z ∈ Γ ± to the orbit γ z with period T z >
0. Choose any f ∈ T and denote f ± := lim r →±∞ f ( r ) ∈ [ − , d ( e f ( r ) α ) tames J ∈ J ( α ), Stokes’ theoremgives(10.1) 0 ≤ E ( u ) = e f + X z ∈ Γ + T z − e f − X z ∈ Γ − T z , hence Γ + cannot be empty. (cid:3) Remark . The proof via Stokes’ theorem works just as well if instead of R × M , u lives in the completion of an exact symplectic cobordism ( W, ω ) withconcave boundary ( M − , ξ − = ker α − ) and convex boundary ( M + , ξ + = ker α + ). Chris Wendl
Recall that this means ∂W = − M − ⊔ M + , and ω = dλ for a 1-form λ that restrictsto positive contact forms λ | T M ± = α ± . As in Lecture 1, we will write J ( W, ω, α + , α − ) ⊂ J ( c W )for the space of almost complex structures J on c W := (( −∞ , × M − ) ∪ M − W ∪ M + ([0 , ∞ ) × M + ) that are compatible with ω on W and belong to J ( α ± ) on the cylin-drical ends. The energy of a J -holomorphic curve u : ( ˙Σ , j ) → ( c W , J ) is then E ( u ) := sup f ∈T Z ˙Σ u ∗ dλ f , where T := { f ∈ C ∞ ( R , ( − , | f ′ > f ( r ) = r near r = 0 } and λ f := e f ( r ) α + on [0 , ∞ ) × M + ,λ on W ,e f ( r ) α − on ( −∞ , × M − . The above proof now generalizes verbatim to show that u must always have a positivepuncture. Notice that in both settings, the argument also gives a uniform boundfor the energy in terms of the periods of the positive asymptotic orbits. Remark . We can also prove Prop. 10.9 using the fact that u ∗ dα ≥ u : ( ˙Σ , j ) → ( R × M, J ) with J ∈ J ( α ). Indeed, Stokes’ theorem then gives(10.2) 0 ≤ Z ˙Σ u ∗ dα = X z ∈ Γ + T z − X z ∈ Γ − T z . The quantity R ˙Σ u ∗ dα is sometimes called the contact area of u . This version ofthe argument however does not easily generalize to arbitrary exact cobordisms.The second proof is based on the maximum principle for subharmonic functions. Proposition . Suppose J ∈ J ( α ) and u = ( u R , u M ) : ( ˙Σ , j ) → ( R × M, J ) is J -holomorphic, where ˙Σ has no boundary. Then u R : ˙Σ → R has no local maxima. Proof.
In any local holomorphic coordinates ( s, t ) on a region in ˙Σ, the non-linear Cauchy-Riemann equation for u is equivalent to the system of equations ∂ s u R − α ( ∂ t u M ) = 0 ,∂ t u R + α ( ∂ s u M ) = 0 ,π ξ ∂ s u M + J π ξ ∂ t u M = 0 , where π ξ : T M → ξ denotes the projection along the Reeb vector field. This gives − ∆ u R = − ∂ s u R − ∂ t u R = − ∂ s [ α ( ∂ t u M )] + ∂ t [ α ( ∂ s u M )]= − dα ( ∂ s u M , ∂ t u M ) = − dα ( π ξ ∂ s u M , J π ξ ∂ s u M ) ≤ J | ξ is tamed by dα | ξ , hence u R is subharmonic. The result thus follows fromthe maximum principle, see e.g. [ Eva98 ]. (cid:3) ectures on Symplectic Field Theory Second proof of Proposition 10.9. If u = ( u R , u M ) : ˙Σ → R × M hasno positive puncture then u R : ˙Σ → R is a proper function bounded above, andtherefore has a local maximum, contradicting Proposition 10.12. (cid:3) Remark . The proof via the maximum principle does not generalize toarbitrary exact cobordisms (
W, dλ ), but it does work in
Stein cobordisms, i.e. if λ f and J are related by λ f = − dF ◦ J for some plurisubharmonic function F : c W → R ,then F ◦ u : ˙Σ → R is subharmonic (cf. [ CE12 ]).With these preliminaries understood, the next two exercises reveal one naturalsetting in which breaking of cylinders can be kept under control. Both exercises areessentially combinatorial.
Exercise . Suppose u is a stable J -holomorphic building in a completedsymplectic cobordism c W with the following properties:(1) u has arithmetic genus 0 and exactly one positive puncture;(2) every connected component of u has at least one positive puncture.Show that u has no nodes, and all of its connected components have exactly onepositive puncture. Exercise . Suppose that in addition to the conditions of Exercise 10.14, u has exactly one negative puncture and no connected component of u is a plane.Show that every level of u then consists of a single cylinder with one positive andone negative end.Exercise 10.15 makes it reasonable to define a Floer-type theory counting onlycylinders in any setting where planes can be excluded, for instance because the Reebvector field has no contractible orbits. This is not always possible, e.g. Hofer [ Hof93 ]proved that on overtwisted contact manifolds, there is always a plane (which is whythe Weinstein conjecture holds). So the invariant we construct will not be definedin such settings, but it happens to be ideally suited to the study of ( T , ξ k ). Fix a closed contact manifold(
M, ξ ) of dimension 2 n − h ∈ [ S , M ]. By primitive , we mean that h is not equal to N h ′ for any h ′ ∈ [ S , M ] and an integer N >
1, and this assumption will be crucial for technical reasons in the following. Given a contact form α for ξ , let P h ( α )denote the set of closed Reeb orbits homotopic to h , where two Reeb orbits areidentified if they differ only by parametrization. Definition . Given a contact manifold (
M, ξ ) and a primitive homotopyclass h ∈ [ S , M ], we will say that a contact form α for ξ is h -admissible if:(1) All orbits in P h ( α ) are nondegenerate; It is to be expected that cylindrical contact homology can be defined also for non-primitivehomotopy classes, but this would require more sophisticated methods to address transversalityproblems. The assumption that h is primitive allows us to assume that all holomorphic curves inthe discussion are somewhere injective, hence they are always regular if J is generic. Chris Wendl (2) There are no contractible closed Reeb orbits.Similarly, we will say that (
M, ξ ) is h -admissible if a contact form with the aboveproperties exists. Definition . Given h ∈ [ S , M ] and an h -admissible contact form α on( M, ξ ), we will say that an almost complex structure J ∈ J ( α ) is h -regular if every J -holomorphic cylinder in R × M with a positive and a negative end both asymptoticto orbits in P h ( α ) is Fredholm regular. Proposition . If h ∈ [ S , M ] is a primitive homotopy class of loops and α is h -admissible on ( M, ξ ) , then the space of h -regular almost complex structuresis comeager in J ( α ) . Proof.
Since h is primitive, the asymptotic orbits for the relevant holomorphiccylinders cannot be multiply covered, hence all of these cylinders are somewhereinjective. The result therefore follows from the standard transversality results provedin Lecture 8 for somewhere injective curves in symplectizations. (cid:3) Proposition . Given an h -admissible contact form α , an h -regular almostcomplex structure J ∈ J ( α ) and an orbit γ ∈ P h ( α ) , suppose u k is a sequence of J -holomorphic cylinders in R × M with one positive puncture at γ and one negativepuncture. Then u k has a subsequence convergent in the SFT topology to a broken J -holomorphic cylinder, i.e. a stable building u ∞ whose levels u ∞ , . . . , u N + ∞ are eachcylinders with one positive and one negative puncture. Moreover, each level satisfies ind( u N ∞ ) ≥ , thus for large k in the convergent subsequence, ind( u k ) = N + X N =1 ind( u N ∞ ) ≥ N + . Proof.
Let’s start with some bad news: the standard SFT compactness the-orem is not applicable in this situation, because we have not assumed that α isnondegenerate, nor even Morse Bott—there is no assumption at all about Reeb or-bits in homotopy classes other than h and 0. This fairly loose set of hypotheses isvery convenient in applications, as nondegeneracy of a contact form is generally aquite difficult condition to check. The price we pay is that we will have to provecompactness manually instead of applying the big theorem (see Remark 10.20). For-tunately, it is not that hard: the crucial point is that in the situation at hand, therecan be no bubbling at all.Indeed, we claim that the given sequence u k : ( R × S , i ) → ( R × M, J ) mustsatisfy a uniform bound | du k | ≤ C with respect to any translation-invariant Riemannian metrics on R × S and R × M .To see this, note first that since all the u k have the same positive asymptotic orbit γ ,their energies are uniformly bounded via (10.1). Thus if | du k ( z k ) | → ∞ for somesequence z k ∈ R × S , we can perform the usual rescaling trick from Lecture 9 anddeduce the existence of a nonconstant finite-energy plane v ∞ : C → R × M . Itssingularity at ∞ cannot be removable since this would produce a nonconstant J -holomorphic sphere, violating Proposition 10.9. It follows that v ∞ is asymptotic to a ectures on Symplectic Field Theory Reeb orbit at ∞ , but this is also impossible since α does not admit any contractibleorbits, and the claim is thus proved.Suppose now that γ has period T + >
0, and observe that by nondegeneracy, theset P h ( α, T + ) := (cid:8) γ ∈ P h ( α ) (cid:12)(cid:12) γ has period at most T + (cid:9) is finite. Let A h ( α ) , A h ( α, T + ) ⊂ (0 , ∞ )denote the set of all periods of orbits in P h ( α ) and P h ( α, T + ) respectively. By(10.2), the negative asymptotic orbit of each u k is in P h ( α, T + ), so we can take asubsequence and assume that these are all the same orbit; call it γ − ∈ P h ( α, T + )and its period T − ∈ A h ( α, T + ). If T − = T + then u ∗ k dα ≡ k , implying thatall u k are the trivial cylinder over γ and thus trivially converge. Assume therefore T − < T + . Then since u ∗ k dα ≥
0, Stokes’ theorem implies that for each k , the function R → R : s Z S u k ( s, · ) ∗ α is increasing and is a surjective map onto ( T − , T + ). The uniform bound on thederivatives implies that for any sequences s k , r k ∈ R with u k ( s k , ∈ { r k } × M , thesequence v k : R × S → R × M : ( s, t ) τ − r k ◦ u k ( s + s k , t )has a subsequence convergent in C ∞ loc ( R × S ) to some finite-energy J -holomorphiccylinder v ∞ : R × S → R × M, which necessarily satisfies Z S v ∞ ( s, · ) ∗ α = lim k →∞ Z S u k ( s + s k , · ) ∗ α ∈ [ T − , T + ]for every s ∈ R . This proves that v ∞ is nonconstant, with a positive puncture at s = ∞ and negative puncture at s = −∞ , and both of its asymptotic orbits arein P h ( α, T + ). If v ∞ is not a trivial cylinder, then it therefore satisfies Z R × S v ∗∞ dα ≥ δ, where δ is any positive number less than the smallest distance between neighboringelements of A h ( α, T + ).Let us call a sequence s k ∈ R nontrivial whenever the limiting cylinder v ∞ obtained by the above procedure is not a trivial cylinder, and call two such sequences s k and s ′ k compatible if s k − s ′ k is not bounded. We claim now that if s k , . . . , s mk is acollection of nontrivial sequences that are all compatible with each other, then m < T + − T − ) δ . Recall from Lecture 9 that we denote the R -translation action on R × M by τ c ( r, x ) := ( r + c, x ). For an alternative argument that v ∞ must have a positive puncture at s = ∞ and negativeat s = −∞ , see Figure 10.3. Chris Wendl
Indeed, we can assume after ordering our collection appropriately and restrictingto a subsequence that s N +1 k − s Nk → ∞ for each N = 1 , . . . , m −
1, and let v N ∞ : R × S → R × M denote the limits of the corresponding convergent subsequences.Then we can find R > Z [ − R,R ] × S ( v N ∞ ) ∗ dα > δ Z [ s Nk − R,s Nk + R ] × S u ∗ k dα > δ N = 1 , . . . , m for sufficiently large k . But these domains are also all disjointfor sufficiently large k , implying T + − T − = Z R × S u ∗ k dα ≥ m X N =1 Z [ s Nk − R,s Nk + R ] × S u ∗ k dα > δm . We’ve shown that there exists a maximal collection of nontrivial sequences s k , . . . , s N + k ∈ R satisfying s N +1 k − s Nk → ∞ for each N , such that if u k ( s Nk , ∈{ r Nk } × M , then after restricting to a subsequence, the cylinders v Nk ( s, t ) := τ − r Nk ◦ u k ( s + s Nk , t )each converge in C ∞ loc ( R × S ) as k → ∞ to a nontrivial J -holomorphic cylinder u N ∞ : R × S → R × M . Let γ ± N denote the asymptotic orbit of u N ∞ at s = ±∞ . Weclaim, γ + N = γ − N +1 for each N = 1 , . . . , N + − . If γ + N = γ − N +1 for some N , choose a neighborhood U ⊂ M of the image of γ + N thatdoes not intersect any other orbit in P h ( α, T + ). Then since each u k is continuous,there must exist a sequence s ′ k ∈ R with s ′ k − s Nk → ∞ and s N +1 k − s ′ k → ∞ such that u k ( s ′ k ,
0) lies in U for all k but stays a positive distance away from theimage of γ + N . A subsequence of ( s, t ) u k ( s + s ′ k , t ) then converges after suitable R -translations to a cylinder u ′∞ : R × S → R × M that cannot be trivial since u ′∞ (0 ,
0) is not contained in any orbit in P h ( α, T + ). This contradicts the assumptionthat our collection s k , . . . , s N + k is maximal. A similar argument shows γ − = γ − and γ + N + = γ, so the curves u ∞ , . . . , u N + ∞ form the levels of a stable holomorphic building u ∞ . Asimilar argument by contradiction also shows that the sequence u k must convergein the SFT topology to u ∞ .Finally, note that since all the breaking orbits in u ∞ are homotopic to h and J is h -regular, the levels u N ∞ are Fredholm regular. Since all of them also come in 1-parameter families of distinct curves related by the R -action, this implies ind( u N ∞ ) ≥ N = 1 , . . . , N + . (cid:3) ectures on Symplectic Field Theory Figure 10.3.
A degenerating sequence of holomorphic cylinders u k : R × S → R × M cannot have a limiting level with a puncture ofthe “wrong” sign unless u k violates the maximum principle for large k . Remark . Nondegeneracy or Morse-Bott conditions are required for sev-eral reasons in the proof of SFT compactness, and indeed, the theorem is not truein general without some such assumption. One can see this by considering whathappens to a sequence u k of J k -holomorphic curves where J k → J ∞ is compati-ble with a sequence of nondegenerate contact forms α k converging to one that isonly Morse-Bott. A compactness theorem for this scenario is proved in [ Bou02 ],but it requires more general limiting objects than holomorphic buildings. On theother hand, it is useful for certain kinds of applications to know when one can dowithout nondegeneracy assumptions and prove compactness anyway. There are twomain advantages to knowing that all Reeb orbits are nondegenerate or belong toMorse-Bott families:(1) It implies that the set of all periods of closed orbits, the so-called actionspectrum of α , is a discrete subset of (0 , ∞ ); in fact, for any T >
0, theset of all periods less than T is finite. Using the relations (10.1) and (10.2),this implies lower bounds on the possible energies of limiting componentsand thus helps show that only finitely many such components can arise.(2) Curves asymptotic to nondegenerate or Morse-Bott orbits also satisfy ex-ponential convergence estimates proved in [ HWZ96 , HWZ01 , HWZ96 , Bou02 ], and similar asymptotic estimates yield a result about “long cylin-ders with small area” (see [
HWZ02 ] and [
BEH + , Prop. 5.7]) whichhelps in proving that neighboring levels connect to each other along break-ing orbits.Our situation in Proposition 10.19 was simple enough to avoid using the “longcylinder” lemma, and we did use the discreteness of the action spectrum, but onlyneeded it for orbits in P h ( α ) since we were able to rule out bubbling in the firststep. An alternative would have been to assume that all orbits (in all homotopyclasses) with period up to the period of γ are nondegenerate: then (10.2) implies Chris Wendl that degenerate orbits never play any role in the main arguments of [
BEH + ], sothe big theorem becomes safe to use. We now define a Z -graded chain complex withcoefficients in Z and generators h γ i for γ ∈ P h ( α ), i.e. CC h ∗ ( M, α ) := M γ ∈P h ( α ) Z . The degree of each generator h γ i ∈ CC h ∗ ( M, α ) is defined by |h γ i| = n − µ CZ ( γ ) ∈ Z , where µ CZ ( γ ) ∈ Z denotes the parity of the Conley-Zehnder index with respect toany choice of trivialization. The choice to write n − Z -grading that we will be able to define under suitable assumptions in Lecture 12. Todefine the differential on CC h ∗ ( M, α ), choose an h -regular almost complex structure J ∈ J ( α ). Given Reeb orbits γ + , γ − ∈ P h ( α ) and a number I ∈ Z , let M I ( J, γ + , γ − )denote the space of all R -equivalence classes of index I holomorphic cylinders in ( R × M, J ) asymptotic to γ ± at ±∞ , i.e. the union of all components M , ( J, A, γ + , γ − ) / R for which vir-dim M , ( J, A, γ + , γ − ) = I . Since J is h -regular, all the curves in M I ( J, γ + , γ − ) are Fredholm regular, so if I ≥ M I ( J, γ + , γ − ) is a smooth mani-fold with dim M I ( J, γ + , γ − ) = I − . Similarly, M ( J, γ + , γ − ) only contains trivial cylinders and is thus empty unless γ + = γ − , and M I ( J, γ + , γ − ) is always empty for I <
0. In particular, M ( J, γ + , γ − )is a discrete set whenever γ + = γ − , and by Proposition 10.19, it is also compact,hence finite. We can therefore define ∂ h γ i = X γ ′ ∈P h ( α ) M ( J, γ, γ ′ ) h γ ′ i , where for any set X , we denote by X the cardinality of X modulo 2. The operator ∂ has odd degree with respect to the grading since every index 1 holomorphic cylinder u with asymptotic orbits γ + and γ − satisfiesind( u ) = 1 = µ τ CZ ( γ + ) − µ τ CZ ( γ − )for suitable choices of the trivialization τ . Following the standard Floer theoretic prescription,the relation ∂ = 0 should arise by viewing the compactification M ( J, γ + , γ − ) foreach γ + , γ − ∈ P h ( α ) as a compact 1-manifold whose boundary is identified with theset of rigid broken cylinders, as these are what is counted by ∂ . Here M ( J, γ + , γ − ) ectures on Symplectic Field Theory is defined as the closure of M ( J, γ + , γ − ) in the space of all J -holomorphic buildingsin R × M modulo R -translation. Proposition 10.19 gives a natural inclusion M ( J, γ + , γ − ) \ M ( J, γ + , γ − ) ⊂ G γ ∈P h ( α ) M ( J, γ + , γ ) × M ( J, γ , γ − ) . We therefore need an inclusion in the other direction, and for this we need to saya word about gluing. We have not had time to discuss gluing in earnest in thesenotes, and we will not do so now either, but the basic idea should be familiar fromFloer homology: given u + ∈ M ( J, γ + , γ ) and u − ∈ M ( J, γ , γ − ), one wouldlike to show that there exists a unique (up to R -translation) one-parameter family { u R ∈ M ( J, γ + , γ − ) } R ∈ [ R , ∞ ) such that u R converges as R → ∞ to the building u ∞ with bottom level u − and top level u + . One starts by constructing a family of preglued maps ˜ u R : R × S → R × M, meaning a smooth family of maps which converge in the SFT topology as R → ∞ to u ∞ but are only approximately J -holomorphic. More precisely, fix parametrizationsof u − and u + and a parametrization of the orbit γ : R /T Z → M such that u + ( s, t ) = exp ( T s,γ ( T t )) h + ( s, t ) for s ≪ ,u − ( s, t ) = exp ( T s,γ ( T t )) h − ( s, t ) for s ≫ , where h ± are vector fields along the trivial cylinder satisfying lim s →∓∞ h ± ( s, t ) = 0.By interpolating between suitable reparametrizations of h + and h − , one can nowdefine ˜ u R such that˜ u R ( s, t ) = τ RT ◦ u + ( s − R, t ) for s ≥ R, ˜ u R ( s, t ) ≈ ( T s, γ ( T t )) for s ∈ [ − R, R ],˜ u R ( s, t ) = τ − RT ◦ u − ( s + 2 R, t ) for s ≤ − R, ¯ ∂ J ˜ u R → R → ∞ . Given regularity of u + and u − , one can now use a quantitative version of the implicitfunction theorem (cf. [ MS04 , § J -holomorphiccylinder u R close to ˜ u R exists for all R sufficiently large. For a more detailed synopsisof the analysis involved, see [ Nel13 , Chapter 7], and [
AD14 , Chapters 9 and 13]for the analogous story in Floer homology. The result is:
Proposition . For an h -admissible α , an h -regular J ∈ J ( α ) and any twoorbits γ + , γ − ∈ P h ( α ) , the space M ( J, γ + , γ − ) admits the structure of a compact -dimensional manifold with boundary, where its boundary points can be identifiednaturally with F γ ∈P h ( α ) M ( J, γ + , γ ) × M ( J, γ , γ − ) . (cid:3) Corollary . The homomorphism ∂ : CC h ∗ ( M, α ) → CC h ∗− ( M, α ) satis-fies ∂ = 0 . (cid:3) We shall denote the homology of this chain complex by HC h ∗ ( M, α, J ) := H ∗ (cid:0) CC h ∗ ( M, α ) , ∂ (cid:1) . Chris Wendl
The goal of the rest of this section is to prove that up to natural isomorphisms, HC h ∗ ( M, α, J ) depends on (
M, ξ ) and h but not on the auxiliary data α and J . For any constant c >
0, there is an obvious bijectionbetween the generators of CC h ∗ ( M, α ) and CC h ∗ ( M, cα ), as the rescaling changesperiods of orbits but not the set of closed orbits itself. Moreover, if J ∈ J ( α ) and J c ∈ J ( cα ) are defined to match on ξ , then there is a biholomorphic diffeomorphism( R × M, J ) → ( R × M, J c ) : ( r, x ) ( cr, x ) , thus giving a bijective correspondence between the moduli spaces of J -holomorphicand J c -holomorphic curves. It follows that our bijection of chain complexes is alsoa chain map and therefore defines a canonical isomorphism(10.3) HC h ∗ ( M, α, J ) = HC h ∗ ( M, cα, J c ) . Next suppose α − and α + are two distinct contact forms for ξ , hence α ± = e f ± α for some fixed contact form α and a pair of smooth functions f ± : M → R . Afterrescaling α + by a constant, we are free to assume f + > f − everywhere. Fix h -regularalmost complex structures J ± ∈ J ( α ± ) and let ∂ ± : CC h ∗ ( M, α ± ) → CC h ∗− ( M, α ± )denote the resulting differentials on the two chain complexes. The region W := (cid:8) ( r, x ) ∈ R × M (cid:12)(cid:12) f − ( x ) ≤ r ≤ f + ( x ) (cid:9) now defines an exact symplectic cobordism from ( M, ξ ) to itself: more precisely,setting M ± := (cid:8) ( f ± ( x ) , x ) ∈ W (cid:12)(cid:12) x ∈ M (cid:9) gives ∂W = − M − ⊔ M + , and the Liouville form λ := e r α satisfies λ | T M ± = α ± .Choose a generic dλ -compatible almost complex structure J on the completion c W that restricts to J ± on the cylindrical ends. Now given γ + ∈ P h ( α + ) and γ − ∈P h ( α − ) and a number I ∈ Z , we shall denote by M I ( J, γ + , γ − )the union of all components M , ( J, A, γ + , γ − ) that have virtual dimension I . Notethat we are not dividing by any R -action here since J need not be R -invariant.Since γ ± are still guaranteed to be simply covered, curves in M I ( J, γ + , γ − ) areagain always somewhere injective and therefore regular, hence M I ( J, γ + , γ − ) is asmooth manifold with dim M I ( J, γ + , γ − ) = I if I ≥
0, and M I ( J, γ + , γ − ) = ∅ for I <
0. The compactification M I ( J, γ + , γ − ) isdescribed via the following straightforward generalization of Proposition 10.19: ectures on Symplectic Field Theory Proposition . For J as described above, suppose u k is a sequence of J -holomorphic cylinders in c W with one positive puncture at an orbit γ ∈ P h ( α + ) andone negative puncture. Then u k has a subsequence convergent in the SFT topologyto a broken J -holomorphic cylinder, i.e. a stable building u ∞ whose levels u N ∞ for N = − N − , . . . , − , , , . . . , N + are each cylinders with one positive and one negativepuncture, living in R × M ± for ± N > and c W for N = 0 . Moreover, the levelssatisfy ind( u ∞ ) ≥ and ind( u N ∞ ) ≥ for N = 0 , thus for large k in the convergentsubsequence, ind( u k ) = N + X N = − N − ind( u N ∞ ) ≥ N − + N + . (cid:3) It follows that the set M ( J, γ + , γ − ) is always finite, and we use this to define amap Φ J : CC h ∗ ( M, α + ) → CC h ∗ ( M, α − ) : h γ i 7→ X γ ′ ∈P h ( α − ) M ( J, γ, γ ′ ) h γ ′ i . This map preserves degrees since it counts index 0 curves, and we claim that it is achain map: Φ J ◦ ∂ + = ∂ − ◦ Φ J . This follows from the fact that by Proposition 10.23 (in conjunction with a corre-sponding gluing theorem), M ( J, γ + , γ − ) is a compact 1-manifold whose boundaryconsists of two types of broken cylinders, depending whether the index 1 curveappears in an upper or lower level: ∂ M ( J, γ + , γ − ) = G γ ∈P h ( α + ) (cid:0) M ( J + , γ + , γ ) × M ( J, γ , γ − ) (cid:1) ∪ G γ ∈P h ( α − ) (cid:0) M ( J, γ + , γ ) × M ( J − , γ , γ − ) (cid:1) . Counting broken cylinders of the first type gives the coefficient in front of h γ − i inΦ J ◦ ∂ + ( h γ + i ), and the second type gives ∂ − ◦ Φ J ( h γ + i ).It follows that Φ J descends to a homomorphism(10.4) Φ J : HC h ∗ ( M, α + , J + ) → HC h ∗ ( M, α − , J − ) . We claim that the map Φ J in (10.4) does notdepend on J . To see this, suppose J and J are two generic choices of compatiblealmost complex structures on c W that both match J ± on the cylindrical ends. Thespace of almost complex structures with these properties is contractible, so we canfind a smooth path { J s } s ∈ [0 , connecting them. For I ∈ Z , consider the parametric moduli space M I ( { J s } , γ + , γ − ) := (cid:8) ( s, u ) (cid:12)(cid:12) s ∈ [0 , , u ∈ M I ( J s , γ + , γ − ) (cid:9) . Chris Wendl
As we observed in Remark 7.4, a generic choice of the homotopy { J s } makes M I ( { J s } ) a smooth manifold withdim M I ( { J s } , γ + , γ − ) = I + 1whenever I ≥ −
1, and M I ( { J s } , γ + , γ − ) = ∅ when I < −
1. Adapting Proposi-tion 10.23 to allow for a converging sequence of almost complex structures, it im-plies that M − ( { J s } , γ + , γ − ) is compact and thus finite, so we can use it to definea homomorphism of odd degree by H : CC h ∗ ( M, α + ) → CC h ∗ +1 ( M, α − ) : h γ i 7→ X γ ′ ∈P h ( α − ) M − ( { J s } , γ, γ ′ ) h γ ′ i . We claim that this is a chain homotopy between Φ J and Φ J , i.e.Φ J − Φ J = ∂ − ◦ H + H ◦ ∂ + . This follows by looking at the boundary of the compactified 1-dimensional space M ( { J s } , γ + , γ − ), which consists of four types of objects:(1) Pairs (0 , u ) with u ∈ M ( J , γ + , γ − ), which are counted by Φ J .(2) Pairs (1 , u ) with u ∈ M ( J , γ + , γ − ), which are counted by Φ J .(3) Pairs ( s, u ) with u a broken cylinder with upper level u + ∈ M ( J + , γ + , γ )and main level u ∈ M − ( J s , γ , γ − ) for some s ∈ (0 , H ◦ ∂ + .(4) Pairs ( s, u ) with u a broken cylinder with lower level u − ∈ M ( J − , γ , γ − )and main level u ∈ M − ( J s , γ + , γ ) for some s ∈ (0 , ∂ − ◦ H .The sum Φ J + Φ J + ∂ − ◦ H + H ◦ ∂ + therefore counts (modulo 2) the boundarypoints of a compact 1-manifold, so it vanishes.Since the action of Φ J on homology no longer depends on J , we will denote itfrom now on by Φ : HC h ∗ ( M, α + , J + ) → HC h ∗ ( M, α − , J − ) . It is well defined for any pair of h -admissible contact forms α ± and h -regular J ± ∈J ( α ± ) since one can first rescale α + to assume α ± = e f ± α with f + > f − , using thecanonical isomorphism (10.3). We claim that for any h -admissible α and h -regular J ∈ J ( α ), the cobordism mapΦ : HC h ∗ ( M, α, J ) → HC h ∗ ( M, α, J )is the identity. Indeed, the literal meaning of this statement is that for any c > HC h ∗ ( M, cα, J c ) → HC h ∗ ( M, α, J )defined by counting index 0 cylinders in a trivial cobordism from (
M, α, J ) to(
M, cα, J c ) is the identity. Writing c = e a for a >
0, the Liouville cobordism inquestion is simply (
W, dλ ) = ([0 , a ] × M, d ( e r α )) , ectures on Symplectic Field Theory and one can choose a compatible almost complex structure on this which matches J and J c on ξ while taking ∂ r to g ( r ) R α for a suitable function g with g ( r ) = 1near r = 0 and g ( r ) = 1 /c near r = a . The resulting almost complex manifoldis biholomorphically diffeomorphic to the usual symplectization ( R × M, J ), so ourcount of index 0 cylinders is equivalent to the count of such cylinders in ( R × M, J ).The latter are simply the trivial cylinders, all of which are Fredholm regular, socounting these defines the identity map on the chain complex.Finally, we need to show that for any three h -admissible pairs ( α i , J i ) with i =0 , ,
2, the cobordism maps Φ ij : HC h ∗ ( M, α j , J j ) → HC h ∗ ( M, α i , J i ) satisfy(10.5) Φ ◦ Φ = Φ . We will only sketch this part: the idea is to use a stretching construction. Afterrescaling, suppose without loss of generality that α i = e f i α with f > f > f . Thenthe cobordism W := (cid:8) ( r, x ) (cid:12)(cid:12) f ( x ) ≤ r ≤ f ( x ) (cid:9) contains a contact-type hypersurface M := (cid:8) ( f ( x ) , x ) (cid:12)(cid:12) x ∈ M (cid:9) ⊂ W . As described at the end of Lecture 9, one can now choose a sequence of compatiblealmost complex structures { J N } N ∈ N on c W that are fixed outside a neighborhoodof M but degenerate in this neighborhood as N → ∞ , equivalent to replacing asmall tubular neighborhood of M with increasingly large collars [ − N, N ] × M inwhich J N belongs to J ( α ). The resulting chain mapsΦ J N : CC h ∗ ( M, α , J ) → CC h ∗ ( M, α , J )are chain homotopic for all N , but as N → ∞ , the index 0 cylinders counted bythese maps converge to buildings with two levels, the top one an index 0 cylinderin the completion of a cobordism from ( M, α , J ) to ( M, α , J ), while the bottomone also has index 0 and lives in a cobordism from ( M, α , J ) to ( M, α , J ). Thecomposition Φ ◦ Φ counts these broken cylinders, so this proves (10.5).In particular, we conclude now that each of the cobordism mapsΦ : HC h ∗ ( M, α + , J + ) → HC h ∗ ( M, α − , J − )is an isomorphism, since composing it with a cobordism map in the opposite di-rection must give the identity. The isomorphism class of HC h ∗ ( M, α, J ) is thereforeindependent of the auxiliary data ( α, J ), and will be denoted by HC h ∗ ( M, ξ ) . This is the cylindrical contact homology of (
M, ξ ) in the homotopy class h . Itis defined for any primitive homotopy class h ∈ [ S , M ] and closed contact manifoldthat is h -admissible in the sense of Definition 10.16. It is also invariant undercontactomorphisms in the following sense: Proposition . Suppose ϕ : ( M , ξ ) → ( M , ξ ) is a contactomorphismwith ϕ ∗ h = h , where h ∈ [ S , M ] is a primitive homotopy class of loops, and ( M , ξ ) is h -admissible. Then ( M , ξ ) is h -admissible, and HC h ∗ ( M , ξ ) ∼ = HC h ∗ ( M , ξ ) . Chris Wendl
Proof.
Given an h -admissible contact form α on ( M , ξ ) and an h -regular J ∈ J ( α ), the contact form α := ϕ ∗ α on M is h -admissible since ϕ defines abijection from P h ( α ) to P h ( α ) and also a bijection between the sets of contractibleReeb orbits for α and α . Since ϕ ∗ ξ = ξ , α is a contact form for ( M , ξ ), hencethe latter is h -admissible. The diffeomorphism ˜ ϕ := Id × ϕ : R × M → R × M then maps ∂ r to ∂ r , R α to R α and ξ to ξ , thus J := ˜ ϕ ∗ J ∈ J ( α ), so ˜ ϕ definesa biholomorphic map ( R × M , J ) → ( R × M , J ) and thus a bijection betweenthe sets of holomorphic cylinders in each. It follows that J is h -regular, and thebijection P h ( α ) → P h ( α ) defines an isomorphism between the chain complexesdefining HC h ∗ ( M , α , J ) and HC h ∗ ( M , α , J ). (cid:3) HC ∗ ( T , ξ k ) The contact form α k on T defined at thebeginning of this lecture has Reeb vector field R k ( ρ, φ, θ ) = cos(2 πkρ ) ∂ θ + sin(2 πkρ ) ∂ φ . Its Reeb orbits therefore preserve and define linear foliations on each of the tori { ρ } × T . In particular, none of the closed orbits are contractible, though all ofthem are also degenerate, as they all come in S -parametrized families foliating { const } × T . For certain homotopy classes h ∈ [ S , T ], this yields a very easycomputation of HC h ∗ ( T , ξ k ), namely whenever h contains no periodic orbits: Theorem . Suppose h ∈ [ S , T ] is any primitive homotopy class of loopssuch that the projection p : T → S : ( ρ, φ, θ ) ρ satisfies p ∗ h = 0 ∈ [ S , S ] . Then α k is h -admissible and the resulting contact homology HC h ∗ ( T , ξ k ) is trivial. (cid:3) Now for the interesting part. Every primitive class h ∈ [ S , T ] not coveredby Theorem 10.25 contains closed orbits of R k , all of them degenerate since theycome in S -parametrized families foliating the tori { const } × T . This makes it notimmediately clear whether ( T , ξ k ) is h -admissible, though the following observationin conjunction with Proposition 10.24 shows that if HC h ∗ ( T , ξ k ) can be defined, itwill be the same for all the homotopy classes under consideration. Lemma . Suppose h , h ∈ [ S , T ] are primitive homotopy classes that areboth mapped to the trivial class under the projection T → S : ( ρ, φ, θ ) ρ . Thenthere exists a contactomorphism ϕ : ( T , ξ k ) → ( T , ξ k ) satisfying ϕ ∗ h = h . Proof.
We can represent h i for i = 0 , γ i ( t ) = (0 , β i ( t )) ∈ S × T , where the loops β i : S → T are embedded and thus represent generatorsof π ( T ) = Z . One can thus find a matrix (cid:18) m np q (cid:19) ∈ SL(2 , Z ) such that thediffeomorphism ϕ : T → T : ( ρ, φ, θ ) ( ρ, mφ + nθ, pφ + qθ )satisfies ϕ ∗ h = h . We have ϕ ∗ α k = [ q cos(2 πkρ ) + n sin(2 πkρ )] dθ + [ p cos(2 πkρ ) + m sin(2 πkρ )] dφ =: F ( ρ ) dθ + G ( ρ ) dφ. ectures on Symplectic Field Theory The loop (
F, G ) : S → R satisfies (cid:18) F ( ρ ) G ( ρ ) (cid:19) = (cid:18) q np m (cid:19) (cid:18) cos(2 πkρ )sin(2 πkρ ) (cid:19) , where (cid:18) q np m (cid:19) ∈ SL(2 , Z ), thus ( F, G ) winds k times about the origin. Any choiceof homotopy from ( F, G ) to (cos(2 πkρ ) , sin(2 πkρ )) through loops ( F s , G s ) : S → R winding k times about the origin with positive rotational velocity then gives rise toa homotopy from ϕ ∗ α k to α k through contact forms F s ( ρ ) dθ + G s ( ρ ) dφ . Gray’sstability theorem therefore yields a contactomorphism ψ : ( T , ξ k ) → ( T , ker ϕ ∗ α k )with ψ smoothly isotopic to the identity. The map ϕ ◦ ψ is thus a contactomorphismof ( T , ξ k ) with ( ϕ ◦ ψ ) ∗ h = ϕ ∗ ψ ∗ h = ϕ ∗ h = h . (cid:3) In light of the lemma, we are free from now on to restrict our attention to theparticular homotopy class h := [ t (0 , , t )] , which is the homotopy class of the 1-periodic orbits foliating the k tori T m := { m/k } × T , m = 0 , . . . , k − R k ( m/k, φ, θ ) = ∂ θ . Though the orbits on these tori are degenerate, it is nothard to show that they all satisfy the Morse-Bott condition; in fact, α k is a Morse-Bott contact form. We will explain a self-contained computation of HC h ∗ ( T , ξ k ) inthe next two sections without using the Morse-Bott condition—but first, it seemsworthwhile to sketch how one can guess the answer using Morse-Bott data.Bourgeois’s thesis [ Bou02 ] gives a prescription for calculating contact homologyin Morse-Bott settings, i.e. for deducing what orbits and what holomorphic curveswill appear under certain standard ways of perturbing the Morse-Bott contact formto make it nondegenerate. Notice first that the only orbits in P h ( α k ) are the onesthat foliate the k tori T , . . . , T k − , and they all have period 1. By (10.2), it followsthat for any J ∈ J ( α k ), there can be no nontrivial J -holomorphic cylinders connect-ing two orbits in P h ( α k ). This makes the calculation of HC h ∗ ( T , ξ k ) sound trivial,but of course there is more to the story since α k is not admissible; indeed, the chaincomplex CC ∗ ( T , α k ) is not even well defined. The prescription in [ Bou02 ] nowgives the following. Each of the families of orbits in T , . . . , T k − is parametrizedby S , and by a standard perturbation technique, any choice of a Morse function f m : S → R for m = 0 , . . . , k − α ′ k that is C ∞ -close to α k ,matches it outside a neighborhood of T m , but has a nondegenerate Reeb orbit on T m for each critical point of f m , while every other closed orbit in the perturbed regioncan be assumed to have arbitrarily large period. Moreover, there is a correspondingperturbation from J ∈ J ( α k ) to J ′ ∈ J ( α ′ k ) such that every gradient flow line of thefunction f m : S → R gives rise to a J ′ -holomorphic cylinder in R × T connectingthe corresponding nondegenerate Reeb orbits along T m . In the present situation,since no J -holomorphic cylinders of the relevant type exist before the perturbation,the only ones after the perturbation are those that come from gradient flow lines.Now imagine performing a similar perturbation near every T , . . . , T k − , usingMorse functions f , . . . , f k − : S → R that each have exactly two critical points. Chris Wendl
For the perturbed contact form α ′ k , P h ( α ′ k ) now consists of exactly 2 k orbits γ ± , . . . , γ ± k − ∈ P h ( α ′ k ) , where we denote by γ + m and γ − m the orbits on T m corresponding to the maximumand minimum of f m respectively. For the obvious choice of trivialization τ for thecontact bundle along γ ± m , one can relate the Conley-Zehnder indices to the Morseindices of the corresponding critical points, giving µ τ CZ ( γ + m ) = 0 , µ τ CZ ( γ − m ) = 1 , m = 0 , . . . , k − . Moreover, the two gradient flow lines connecting maximum and minimum for each f m give rise two exactly two holomorphic cylinders in M ( J ′ , γ − m , γ + m ) for each m =0 , . . . , k −
1, and these are all the curves that are counted for the differential on CC h ∗ ( T , α ′ k , J ′ ). Counting modulo 2, we thus have ∂ h γ ± m i = 0 for all m = 0 , . . . , k − , implying HC h ∗ ( T , α ′ k , J ′ ) = ( Z k ∗ = odd , Z k ∗ = even . Let us state this as a theorem.
Theorem . Suppose h ∈ [ S , T ] is a primitive homotopy class that mapsto the trivial class under the projection T → S : ( ρ, φ, θ ) ρ . Then ( T , ξ k ) is h -admissible and HC h ∗ ( T , ξ k ) ∼ = ( Z k ∗ = odd , Z k ∗ = even . Theorem 10.1 is an immediate corollary of this: indeed, if ϕ : ( T , ξ k ) → ( T , ξ ℓ )is a contactomorphism, choose any h ∈ [ S , T ] for which Theorem 10.27 applies,and let h := ϕ ∗ h ∈ [ S , T ]. Then HC h ∗ ( T , ξ ℓ ) ∼ = Z ℓ implies via Proposition 10.24that HC h ∗ ( T , ξ k ) ∼ = Z ℓ . But Theorems 10.25 and 10.27 imply that the latter isalso either 0 or Z k , hence k = ℓ . In preparation for giving a self-contained proof of Theorem 10.27, we now explain a general procedure for relatingholomorphic cylinders in a symplectization to solutions of the Floer equation. Thisidea is loosely inspired by arguments in [
EKP06 ].To motivate what follows, notice that on a neighborhood of T = { } × T ⊂ ( T , ξ k ), we can write α k = cos(2 πkρ ) ( dθ + β ) , where β := tan(2 πkρ ) dφ defines a Liouville form on the annulus A := [ − / , / × S with coordinates ( ρ, φ ). This makes the neighborhood A × S ⊂ ( T , ξ k ) a specialcase of the following general construction. ectures on Symplectic Field Theory Definition . Suppose V is a 2 n -dimensional manifold with an exact sym-plectic form dβ . The contact manifold ( V × S , ker( dθ + β )) is then called the contactization of ( V, β ). Here θ denotes the coordinate on the S factor.It’s easy to check that dθ + β is indeed a contact form on V × S whenever dβ is symplectic on V : the latter means ( dβ ) n > V , so( dθ + β ) ∧ [ d ( dθ + β )] n = ( dθ + β ) ∧ ( dβ ) n = dθ ∧ ( dβ ) n > . Now here’s a cute trick one can play with contactizations. For the rest of thissubsection, assume (
V, dβ )is an arbitrary compact 2 n -dimensional exact symplectic manifold with boundary.Fix a smooth function H : V × S → R , which we shall think of in the following as a time-dependent Hamiltonian H θ := H ( · , θ ) : V → R on ( V, dβ ). The 2-form on V × S defined byΩ = dβ + dθ ∧ dH = d ( β − H dθ )is then fiberwise symplectic , meaning its restriction to each of the fibers of theprojection map V × S → S is symplectic. We claim that for every ǫ > λ ǫ := dθ + ǫ ( β − H dθ )defines a contact form on V × S . This is a variation on the construction that wasused by Thurston and Winkelnkemper [ TW75 ] to define contact forms out of openbook decompositions, and the proof is simple enough: since dλ ǫ = ǫ Ω, we just needto check that λ ǫ ∧ Ω n > ǫ > λ ǫ ∧ Ω n = dθ ∧ ( dβ ) n + ǫ ( β − H dθ ) ∧ Ω n > ǫ is small. To see the relation between λ ǫ and the contactization, we can write λ ǫ = (1 − ǫH ) dθ + ǫβ = (1 − ǫH ) (cid:18) dθ + ǫ − ǫH β (cid:19) and observe that ǫ − ǫH β is also a Liouville form on V whenever H is θ -independentand ǫ > R ǫ for λ ǫ vary with ǫ , but their directions do not, since dλ ǫ = ǫ Ω has the same kernel for every ǫ . Moreover, while λ ǫ ceases to be a contactform when ǫ →
0, the Reeb vector fields still have a well-defined limit: they convergeas ǫ → R satisfying dθ ( R ) ≡ R , · ) ≡ . The latter can be written more explicitly as R = ∂ θ + X θ , Elsewhere in the literature, the contactization is also often defined as V × R instead of V × S .The usage here is consistent with [ MNW13 ]. Chris Wendl where X θ is the time-dependent Hamiltonian vector field determined by H θ , i.e. viathe condition dβ ( X θ , · ) = − dH θ . As one can easily compute, the reason for this nice behavior as ǫ → R ǫ are also the Reeb vector fields for a smooth family of stable Hamiltonian structures: Proposition . The pairs H ǫ := (Ω , λ ǫ ) for ǫ ≥ sufficiently small define asmooth family of stable Hamiltonian structures whose Reeb vector fields are R ǫ . (cid:3) We shall write the hyperplane distributions induced by H ǫ asΞ ǫ := ker λ ǫ ⊂ T ( V × S ) . These are contact structures for ǫ > J ( H ǫ ) of R -invariantalmost complex structures on R × ( V × S ) compatible with H ǫ is then identicalto J ( λ ǫ ). On the other hand for ǫ = 0, Ξ = ker dθ is a foliation, namely it is thevertical subbundle of the trivial fibration V × S → S . To interpret H , notice thatits closed Reeb orbits in the homotopy class of γ : S → V × S : t (const , t ) areall of the form γ ( t ) = ( x ( t ) , t ) where x : S → V is a contractible 1-periodic orbitof X θ . Moreover, suppose J ∈ J ( H ), which is equivalent to a choice of compatiblecomplex structure on the symplectic bundle (Ξ , Ω | Ξ ), or in other words, an S -parametrized family of dβ -compatible almost complex structures { J θ } θ ∈ S on V .Then if u = ( f, v, g ) : R × S → R × ( V × S )is a J -holomorphic cylinder asymptotic at {±∞} × S to two orbits of the formdescribed above, the nonlinear Cauchy-Riemann equation for u turns out to implythat ( f, g ) : R × S → R × S is a holomorphic map with degree 1 sending {±∞}× S to {±∞} × S , and we can therefore choose a unique biholomorphic reparametriza-tion of u so that ( f, g ) becomes the identity map. Having done this, the equationsatisfied by v : R × S → V is now ∂ s v + J t ( v )( ∂ t v − X t ( v )) = 0 , in other words, the Floer equation for the data { J θ } θ ∈ S and { H θ } θ ∈ S .To complete the analogy, notice that since Ω is exact, we can write down anatural symplectic action functional with respect to each H ǫ as A ǫ : C ∞ ( S , V × S ) → R : γ Z S γ ∗ ( β − H dθ ) . For loops of the form γ ( t ) = ( x ( t ) , t ) with x : S → V contractible, this reduces(give or take a sign—see Remark 10.32) to the usual formula for the Floer actionfunctional(10.6) A H ( γ ) = Z S x ∗ β − Z S H ( x ( t )) dt = Z D ¯ x ∗ dβ − Z S H ( x ( t )) dt, where ¯ x : D → V is any map satisfying ¯ x | ∂ D = x . Stokes’ theorem gives an easyrelation between the action and the so-called Ω -energy if u : R × S → R × ( V × S ) ectures on Symplectic Field Theory is a J -holomorphic curve for J ∈ J ( H ǫ ) and is positively/negatively asymptotic toorbits γ ± : S → V × S at s = ±∞ : we have0 ≤ Z R × S u ∗ Ω = A ǫ ( γ + ) − A ǫ ( γ − ) . If u ( s, t ) = ( s, v ( s, t ) , t ), then the left hand side is identical to the definition of energyin Floer homology, namely E H ( v ) := Z R × S dβ ( ∂ s v, ∂ t v − X t ( v )) ds ∧ dt = Z R × S dβ ( ∂ s v, J t ( v ) ∂ s v ) ds ∧ dt, thus giving the familiar relation(10.7) E H ( v ) = A H ( γ + ) − A H ( γ − ) . To relate this to the usual notion of energy with respect to a stable Hamiltonianstructure, we write the usual formula E ǫ ( u ) := sup ϕ ∈T Z ˙Σ u ∗ (cid:2) d (cid:0) ϕ ( r ) λ ǫ (cid:1) + Ω (cid:3) , with T := (cid:8) ϕ ∈ C ∞ ( R , ( − ǫ , ǫ )) (cid:12)(cid:12) ϕ ′ > (cid:9) for some constant ǫ > ǫ , Stokes’ theorem gives a bound for E ǫ ( u ) interms of the asymptotic orbits of u since Ω is exact. Finally, in the case ǫ = 0 with u ( s, t ) = ( s, v ( s, t ) , t ), we find E ( u ) = sup ϕ ∈T Z R × S ϕ ′ ( s ) ds ∧ dt + Z R × S u ∗ Ω = 2 ǫ + E H ( v ) , so bounds on E ( u ) are equivalent to bounds on the Floer homological energy E H ( v ).The basic fact that Floer trajectories v : R × S → V with E H ( v ) < ∞ are asymp-totic to contractible 1-periodic Hamiltonian orbits can now be regarded as a corollaryof our Theorem 9.6.The above discussion gives a one-to-one correspondence between a certain mod-uli space of unparametrized J -holomorphic cylinders in R × ( V × S ) and the mod-uli space of Floer trajectories between contractible 1-periodic orbits in ( V, dβ ) withHamiltonian function H . If we can adequately understand the moduli space of Floertrajectories—in particular if we can classify them and prove that they are regular—then the idea will be to extend this classification via the implicit function theorem toany J ǫ ∈ J ( λ ǫ ) sufficiently close to J for ǫ > γ ( t ) = x at critical points x ∈ Crit( H ),and for each such orbit, γ ∗ Ξ has a canonical homotopy class of unitary trivializa-tions, the so-called constant trivialization . The following fundamental result iscommonly used in proving the isomorphism from Hamiltonian Floer homology tosingular homology. Theorem . Suppose H : V → R is a smooth Morse function with nocritical points on the boundary, J is a fixed dβ -compatible almost complex structureon V , and the gradient flow of H with respect to the metric dβ ( · , J · ) is Morse-Smale Chris Wendl and transverse to ∂V . Given δ > , let H δ := δH : V → R , with Hamiltonian vectorfield X H δ = δX H , and consider the stable Hamiltonian structure H δ := ( dβ + dθ ∧ dH δ , dθ ) on V × S with induced Reeb vector field R δ = ∂ θ + X H δ . Then for all δ > sufficiently small, the following statements hold.(1) The -periodic R δ -orbit γ x : S → V × S : t ( x, t ) arising from anycritical point x ∈ Crit( H ) is nondegenerate, and its Conley-Zehnder indexrelative to the constant trivialization τ is related to the Morse index ind( x ) ∈{ , . . . , n } by (10.8) µ τ CZ ( γ x ) = n − ind( x ) . (2) Any trajectory γ : R → V satisfying the negative gradient flow question ˙ γ = −∇ H δ ( γ ) gives rise to a Fredholm regular solution v : R × S → V :( s, t ) γ ( s ) of the time-independent Floer equation (10.9) ∂ s v + J ( v )( ∂ t v − X H δ ( v )) = 0 , and the virtual dimensions of the spaces of Floer trajectories near v andgradient flow trajectories near γ are the same.(3) Every -periodic orbit of X H δ in ˚ V is a constant loop at a critical pointof H .(4) Every finite-energy solution v : R × S → ˚ V of (10.9) is of the form v ( s, t ) = γ ( s ) for some negative gradient flow trajectory γ : R → V . Proof.
The following proof is based on arguments in [
SZ92 ], see in particularTheorem 7.3.For the first statement, let γ ( t ) = ( x, t ) for x ∈ Crit( H ) and recall from Lecture 3the formula for the asymptotic operator of a 1-periodic orbit, A γ : Γ( γ ∗ Ξ ) → Γ( γ ∗ Ξ ) : η
7→ − J (cid:0) ∇ t η − ∇ η R δ (cid:1) , where ∇ is any symmetric connection on V × S . Identifying Γ( γ ∗ Ξ ) in the naturalway with C ∞ ( S , T x V ), using the trivial connection and writing R δ ( z, θ ) = ∂ θ + X H δ ( z ) = ∂ θ + δJ ( z ) ∇ H ( z ), A γ becomes the operator A γ = − J ∂ t − δ ∇ H ( x )on C ∞ ( S , T x V ), where ∇ H ( x ) : T x V → T x V denotes the Hessian of H at x .Choosing a unitary basis for T x V identifies this with − J ∂ t − δS for some symmetric2 n -by-2 n matrix S and the standard complex structure J = (cid:18) − (cid:19) , so ker A γ corresponds to the space of 1-periodic solutions to ˙ η = δJ Sη . The Morse conditionimplies that S is nonsingular, so the eigenvalues of δJ S are all nonzero, but theyare also small since δ is small. It follows that nontrivial solutions of ˙ η = δJ Sη cannot be 1-periodic if S is nonsingular and δ is sufficiently small, thus proving thatker A γ is trivial, hence γ is nondegenerate. ectures on Symplectic Field Theory To calculate µ τ CZ ( γ ), note that λ ∈ σ ( A γ ) if and only if there exists a nontrivial1-periodic solution η to the equation˙ η = J ( δS + λ ) η. If δ and λ are both close to 0, then the same argument again implies that no suchsolution exists unless δS + λ is singular, meaning λ ∈ σ ( − δS ). On the other hand,any constant loop η ( t ) ∈ ker( λ + δS ) furnishes an element of the λ -eigenspace of A γ ,so we obtain a bijection between the spectra of A γ and − δS in some neighborhoodof 0. It follows that if S ± denotes a pair of nonsingular symmetric matrices definingasymptotic operators A ± = − J ∂ t − δS ± , then the spectral flows are related by µ spec ( A − , A + ) = − µ spec ( S − , S + )when δ > S ± by E − ( S ± ), this relation impliesdim E − ( S + ) − dim E − ( S − ) = µ CZ ( A − ) − µ CZ ( A + ) . Now suppose S + is a coordinate expression for the Hessian ∇ H ( x ), hence dim E − ( S + ) =ind( x ) and µ CZ ( A + ) = µ τ CZ ( γ ). Choosing S − = (cid:18) − (cid:19) then gives dim E − ( S − ) = n and µ CZ ( A − ) = 0 by definition, so µ τ CZ ( γ ) = n − ind( x ) follows.The second statement follows in a similar manner by writing down and compar-ing the linearized operators for the Floer equation and the negative gradient flowequation. Let’s leave this as an exercise.For the third statement, suppose we have a sequence δ k → x k : S → ˚ V satisfying ˙ x k = X H δk ( x k ) = δ k X H ( x k ). Pick a number c > δ = c , choose a sequence ofintegers N k ∈ N such that N k δ k → c, and consider the loops y k : S → ˚ V : t x k ( N k t ). These satisfy˙ y k = N k δ k X H ( y k ) , and since X H is C ∞ -bounded on V and N k δ k is also bounded, the Arzel`a-Ascolitheorem provides a subsequence with y k → y ∞ in C ∞ ( S , V ) , where y ∞ : S → V satisfies ˙ y ∞ = X H c ( y ∞ ) for H c := cH : V → R . But y ∞ isalso constant: indeed, since y k ( t + 1 /N k ) = y k ( t ) and N k → ∞ , we can find for any t ∈ S a sequence q k ∈ Z satisfying q k /N k → t , so(10.10) y ∞ ( t ) = lim k →∞ y k ( q k /N k ) = lim k →∞ y k (0) = y ∞ (0) . Since the constant orbit y ∞ is nondegenerate by part (1) of the theorem, there canonly be one sequence of solutions to ˙ y k = X H Nkδk ( y k ) converging to y ∞ , and weconclude that y k is also constant for all k sufficiently large.We will now use a similar trick to prove the fourth statement in the theorem.We shall work under the additional assumption that(10.11) | ind( x ) − ind( y ) | ≤ x, y ∈ Crit( H ) , Chris Wendl which suffices for the application in § Suppose to the contrary that there exists a sequence of positive numbers δ k → v k : R × S → ˚ V of the equation ∂ s v k + J ( v k )( ∂ t v k − X H δk ( v k )) = 0, where each v k ( s, t ) is not t -independent. By part (3) of the theorem,we can restrict to a subsequence and assume each v k for large k is asymptotic toa fixed pair of critical points x ± = lim s →±∞ v k ( s, · ) ∈ Crit( H ), and x + = x − since v k would otherwise by constant and therefore t -independent. Choose a sequence N k ∈ N with N k → ∞ and N k δ k → c, where c > δ = c . Define w k : R × S → V by w k ( s, t ) = v k ( N k s, N k t ) . Then w k satisfies another time-independent Floer equation,(10.12) ∂ s w k + J ( w k ) ( ∂ t w k − X H Nkδk ( w k )) = 0 , where the Hamiltonian functions H N k δ k converge to H c . The standard compactnesstheorem for Floer trajectories should now imply that a subsequence of w k convergesto a broken Floer trajectory whose levels will be t -independent. Since the settingmay seem a bit nonstandard, here are some details.The sequence w k is uniformly C -bounded since V is compact. We claim thatit is also C -bounded. If not, then there is a sequence z k = ( s k , t k ) ∈ R × S with | dw k ( z k ) | =: R k → ∞ , and we can use the usual rescaling trick from Lecture 9 todefine a sequence f k : D ǫ k R k → V : z w k ( z k + z/R k )for a suitable sequence ǫ k → ǫ k R k → ∞ and | dw k ( z ) | ≤ R k for all z ∈ D ǫ k ( z k ). The latter implies that f k satisfies a local C -bound independent of k , andsince ∂ s f k + J ( f k ) (cid:18) ∂ t f k − R k J ( f k ) X H Nkδk ( f k ) (cid:19) , elliptic regularity (see Remark 10.31 below) provides a subsequence for which f k converges in C ∞ loc ( C , V ) to a J -holomorphic plane f ∞ : C → V , which is nonconstantsince | df ∞ (0) | = lim k →∞ | df k (0) | = 1 . Since v k and therefore w k are all asymptotic to fixed constant orbits x ± , we have auniform bound on the Floer energies of w k , E H Nkδk ( w k ) = A H Nkδk ( x + ) − A H Nkδk ( x − ) = N k δ k [ H ( x − ) − H ( x + )] , (10.13) Lifting this assumption requires gluing, whereas we shall only need the usual implicit functiontheorem for Fredholm regular solutions of the Floer equation. ectures on Symplectic Field Theory where the right hand side is bounded since N k δ k → c . Using change of variables andthe fact that dβ ( ∂ s f k , J ( f k ) ∂ s f k ) ≥
0, this implies a uniform bound Z D ǫkRk dβ ( ∂ s f k ,J ( f k ) ∂ s f k ) ds ∧ dt = Z D ǫk ( zk ) dβ ( ∂ s v k , J ( v k ) ∂ s v k ) ds ∧ dt ≤ Z R × S dβ ( ∂ s v k , J ( v k ) ∂ s v k ) ds ∧ dt = E H Nkδk ( w k ) ≤ C, thus Z C f ∗∞ dβ = Z C dβ ( ∂ s f ∞ , ∂ t f ∞ ) ds ∧ dt = Z C dβ ( ∂ s f ∞ , J ( f ∞ ) ∂ s f ∞ ) ds ∧ dt < ∞ . The removable singularity theorem now extends f ∞ to a nonconstant J -holomorphicsphere f ∞ : S → V , but this violates Stokes’ theorem since J is tamed by an exactsymplectic form.We’ve now shown that the sequence w k : R × S → V is uniformly C -bounded,and it has bounded energy due to (10.13). Pick any sequence s k ∈ R and considerthe sequence of translated Floer trajectories˜ w k ( s, t ) := w k ( s + s k , t ) . These are also uniformly C -bounded, so by elliptic regularity (see Remark 10.31again), a subsequence converges in C ∞ loc ( R × S ) to a map w ∞ : R × S → V satisfying ∂ s w ∞ + J ( w ∞ ) ( ∂ t w ∞ − X H c ( w ∞ )) = 0 , and it has finite energy E H c ( w ∞ ) < ∞ due to (10.13), implying that w ∞ is asymp-totic to a pair of 1-periodic orbits of X H c as s → ±∞ . By the same argument usedin (10.10) above, w ∞ is also t -independent. It follows that w ∞ ( s, t ) = γ ∞ ( s ) forsome nonconstant gradient flow trajectory γ ∞ : R → ˚ V . Depending on the choice ofsequence s k , this trajectory may or may not be constant, but we can always choose s k to guarantee that γ ∞ is not constant: indeed, since each w k is asymptotic to twoseparate critical points at ±∞ , s k ∈ R can be chosen such that w k ( s k ,
0) stays afixed distance away from every critical point of H , and then w ∞ (0 ,
0) = lim k →∞ w k ( s k , Crit( H c ) . One can now adapt the argument of Proposition 10.19 to find various sequences s k ∈ R that yield potentially separate limiting trajectories forming the levels of a brokentrajectory, which is the limit of w k in the Floer topology. But since all the levelsare t -independent and the gradient flow of H c is Morse-Smale, condition (10.11)implies that the most complicated (and therefore the only) limit possible involvesa single level w ∞ ( s, t ) = γ ( s ), which is a gradient flow trajectory between criticalpoints whose Morse indices differ by 1. This trajectory is Fredholm regular and hasindex 1 due to part (2) of the theorem, thus by the implicit function theorem, theonly solutions to (10.12) that can converge to w ∞ are the obvious reparametrizationsof γ , i.e. they are also t -independent. This is a contradiction. (cid:3) Remark . In previous lectures we’ve used the theorem that “ C -boundsimply C ∞ -bounds” to prove compactness for J -holomorphic curves, but not for Chris Wendl solutions of inhomogeneous Cauchy-Riemann type equations such as the Floer tra-jectories w k and rescalings f k in the above proof. There is an easy trick to reducethese to our standard setup: as we’ve already seen, solutions of the Floer equationare equivalent to honest pseudoholomorphic curves in the symplectization of a cer-tain stable Hamiltonian structure, which is a manifold of two dimensions higher.A similar trick can be used for any inhomogeneous Cauchy-Riemann type equa-tion ¯ ∂ J f = ν , reducing it to an honest Cauchy-Riemann type equation at the cost ofadding two dimensions. This trick was used already by Gromov, see [ Gro85 , 1.4.C].
Remark . You may notice with some horror that (10.8) differs by a signfrom what is stated in [
SZ92 ]. As far as I can tell, the discrepancy arises fromthe fact that while Floer homology is traditionally defined in terms of a negativegradient flow for the action functional, SFT is based on a positive gradient flow—this is also why the action functional in (10.6) differs by a sign from what we sawin Lecture 1. If one takes as an axiom that the Conley-Zehnder index should serveas a “relative Morse index” for the action functional, then changing the sign of thefunctional also reverses the signs of Conley-Zehnder indices, so as a result thereappear to be two parallel sign conventions for Conley-Zehnder indices in differentsectors of the literature. I’m sorry. It’s not my fault.Returning now to the family H ǫ , choose δ > V × S by H δǫ = (Ω δ , λ δǫ ) , where Ω δ := dβ + dθ ∧ dH δ and λ δǫ := dθ + ǫ ( β − H δ dθ ) . Denote the induced hyperplane distributions and Reeb vector fields by Ξ δǫ and R δǫ respectively. We have only changed the Hamiltonian H by rescaling, so all previousstatements about H ǫ also apply to H δǫ , in particular λ δǫ is contact and J ( H δǫ ) = J ( λ δǫ )for all ǫ > ǫ may now depend on δ . Once δ > ǫ > Theorem . Assume the same hypotheses as in Theorem 10.30, including (10.11) , and denote the unique extension of J to an R -invariant almost complexstructure in J ( H δ ) by J . Given δ sufficiently small and any smooth family ofcompatible R -invariant almost complex structures J ǫ ∈ J ( H δǫ ) matching J at ǫ = 0 ,there exists ǫ > such that every critical point x ∈ Crit( H ) gives rise to a smoothfamily of nondegenerate closed R δǫ -orbits x ǫ : S → V × S ǫ ∈ [0 , ǫ ] with x ( t ) = ( x, t ) , and every gradient flow trajectory γ : R → V for H gives rise toa smooth family of Fredholm regular J ǫ -holomorphic cylinders u ǫγ : R × S → R × ( V × S ) ǫ ∈ [0 , ǫ ] with u γ ( s, t ) = ( s, γ ( δs ) , t ) . Moreover, for all ǫ ∈ [0 , ǫ ] , every closed R δǫ -orbit homo-topic to t (const , t ) belongs to one of the families x ǫ up to parametrization, and ectures on Symplectic Field Theory every J ǫ -holomorphic cylinder with a positive and a negative end asymptotic to orbitsof this type belongs to one of the families u ǫγ , up to biholomorphic parametrization. Proof.
The first part is immediate from the implicit function theorem since theorbits x ( t ) = ( x, t ) are nondegenerate and the curves u γ ( s, t ) = ( s, γ ( δs ) , t ) are Fred-holm regular by Theorem 10.30. For the uniqueness statement, observe that if ǫ k → γ k is a sequence of R δǫ k -orbits in the relevant homotopy class, then their periodsare uniformly bounded, so Arzel`a-Ascoli gives a subsequence convergent to a closed R δ -orbit, which is a nondegenerate orbit of the form x ( t ) = ( x, t ) for x ∈ Crit( H )by Theorem 10.30, so sequences converging to this orbit are unique by the implicitfunction theorem. A similar argument proves uniqueness of J ǫ -holomorphic cylin-ders: if ǫ k → u k is a J ǫ k -holomorphic sequence, then first by the uniquenessof the orbits, we can extract a subsequence for which all u k are asymptotic at bothends to orbits in fixed families x ǫ k ± converging to x ± ( t ) = ( x ± , t ) as k → ∞ . SinceΩ is exact, Stokes’ theorem then gives a uniform bound on the energies E ǫ k ( u k ).Since all R δ -orbits in the relevant homotopy class are nondegenerate and none arecontractible, one can now prove as in Proposition 10.19 that u k has a subsequenceconvergent to a finite-energy stable J -holomorphic building u ∞ consisting only ofcylinders. Its levels are asymptotic to orbits of the form x ( t ) = ( x, t ) for x ∈ Crit( H ),thus they can be parametrized as ( s, t ) ( s, v ( s, t ) , t ) for v : R × S → V satisfyingthe H δ -Floer equation, hence v ( s, t ) = γ ( δs ) by Theorem 10.30. Now since ∇ H isMorse-Smale and indices of critical points can only differ by at most 1, the building u ∞ can have at most one nontrivial level u ∞ ( s, t ) = ( s, γ ( δs ) , t ), implying u k → u ∞ .Since u ∞ is Fredholm regular, the implicit function theorem does the rest. (cid:3) ( T , ξ k ) . We now complete the computation ofthe cylindrical contact homology HC h ∗ ( T , ξ k ). We can assume via Lemma 10.26that h is the homotopy class of the orbits in the special set of tori T m = { m/k } × T ⊂ T , m = 0 , . . . , k − . Let’s focus for now on the case k = 1, as the general case will simply be a k -foldcover of this. Thanks to the Morse-Bott discussion in § h -admissible contact form α for ( T , ξ ) such that P h ( α )contains exactly two orbits, both in T ⊂ T , along with an h -regular J ∈ J ( α ) suchthat the differential on CC h ∗ ( T , α ) counts exactly two J -holomorphic cylinders thatconnect the two orbits in T . Let A denote the annulus A = [ − , × S with coordinates ( ρ, φ ). This will play the role of the Liouville manifold ( V, dβ ) fromthe previous section, and we set β := ρ dφ. For the Hamiltonian H : A → R , choose a Morse function with the followingproperties:(1) H has a minimum at x = (0 , x = (0 , / H ( ρ, φ ) = | ρ | for 1 / ≤ | ρ | ≤ Chris Wendl (3) The gradient flow of H with respect to the standard Euclidean metric on[ − , × S is Morse-Smale.Fix a number δ > H δ := δH in A , and since it will turn out to be useful in Lemma 10.34below, assume without loss of generality δ ∈ Q . Then following the prescription described above, we consider the family of stableHamiltonian structures H δǫ = (Ω δ , λ δǫ ) on A × S for ǫ ≥ λ δǫ = (1 − ǫδH ) dθ + ǫρ dφ, Ω δ = dρ ∧ dφ + δ dθ ∧ dH, with induced Reeb vector fields R δǫ and hyperplane distributions Ξ δǫ := ker λ δǫ .Choose J ǫ ∈ J ( H δǫ ) to be any smooth family such that J | Ξ δ matches the stan-dard complex structure on A defined by J ∂ ρ = ∂ φ . Then for all ǫ > R δǫ -orbitsin A × S homotopic to t (0 , , t ), as well as a classification of all J ǫ -holomorphiccylinders asymptotic to them. Up to parametrization, there are exactly two suchorbits, γ ǫi : S → A × S , i = 0 , , which correspond to the Morse critical points x and x and thus by (10.8) haveConley-Zehnder indices µ τ CZ ( γ ǫi ) = 1 − ind( x i ) = 1 − i ∈ { , } relative to the constant trivialization τ . There are also exactly two J ǫ -holomorphiccylinders u ǫ ± : R × S → R × ( A × S ) , corresponding to the two negative gradient flow lines that descend from x to x ,thus the u ǫ ± are index 1 curves with a negative end approaching γ ǫ and a positiveend approaching γ ǫ . If we can suitably embed this model into ( T , ξ ) and show thatall the orbits and curves needing to be counted are contained in the model, then wewill have a complete description of HC h ∗ ( T , ξ ), with two generators h γ ǫ i and h γ ǫ i ,of even and odd degree respectively, satisfying ∂ h γ ǫ i = 2 h γ ǫ i = 0 and ∂ h γ ǫ i = 0since the former counts two curves and the latter counts none. Lemma . For any ǫ > sufficiently small, there exists a contact embeddingof ( A × S , ker λ δǫ ) ֒ → ( T , ξ ) identifying the homotopy class of the loops t (0 , , t ) in A × S with h . Moreover,the contact form λ δǫ and almost complex structure J ǫ ∈ J ( H δǫ ) can then be extended toan h -admissible contact form α on ( T , ξ ) and an h -regular almost complex structure J ∈ J ( α ) such that γ ǫ and γ ǫ are the only orbits in P h ( α ) , and all J -holomorphiccylinders with a positive and a negative end asymptotic to either of these orbits arecontained in the interior of A × S . ectures on Symplectic Field Theory Proof.
We’ve chosen β and H so that in the region 1 / ≤ | ρ | ≤ α := λ δǫ = (1 − ǫδ | ρ | ) dθ + ǫρ dφ =: f ( ρ ) dθ + g ( ρ ) dφ, so the Reeb vector field on this region has the form D ( ρ ) ( g ′ ( ρ ) ∂ θ − f ′ ( ρ ) ∂ φ ). Noticethat f ′ ( ρ ) g ′ ( ρ ) = ∓ ǫδǫ = ∓ δ, and we assumed δ ∈ Q , so the Reeb orbits in this region are all periodic. Next, picka large number N ≫ α to a contact form on [ − N, N ] × S × S viathe same formula. Now extend the path ( f, g ) : [ − N, N ] → R to R such that it hasperiod 2 N + 2 and winds once around the origin over the interval [ − N − , N + 1],with positive angular velocity. This produces a contact form α on T N := (cid:16) R . (2 N + 2) Z (cid:17) × S × S which takes the form f ( ρ ) dθ + g ( ρ ) dφ outside of | ρ | ≤ /
2. We claim in fact that α is homotopic through contact forms to one that takes this form globally, where( f, g ) may be assumed to be a smooth loop winding once around the origin. To seethis, one need only homotop H in the region | ρ | ≤ / ρ -coordinate; the contact condition holds for all Hamiltoniansin this homotopy as long as ǫ > T N → T : ( ρ, φ, θ ) (cid:18) ρ N + 2 , φ, θ (cid:19) pushes ker α forward to a contact structure isotopic to one of the form F ( ρ ) dθ + G ( ρ ) dφ for a loop ( F, G ) : S → R winding once around the origin, so taking ahomotopy of this loop to (cos(2 πρ ) , sin(2 πρ )) and applying Gray’s stability theoremproduces a contactomorphism ( T N , ker α ) → ( T , ξ )that is isotopic to the above diffeomorphism.The construction clearly guarantees that no closed Reeb orbit of α outside A × S is homotopic to the preferred class h , and there are also no contractible orbits, so α is an h -admissible contact form on T N . Choose any extension of J ǫ to some J ∈ J ( α ) on T N . We claim now that if N is chosen sufficiently large, then no J -holomorphic cylinder in R × T N with one positive end at either of the orbits γ ǫi can ever venture outside the region R × ( − / , / × T . Suppose in particularthat u is such a curve and its image intersects R × { / } × T . Since the entireregion [1 / , N ] × T is foliated by closed Reeb orbits, we can define Υ to be the setof Reeb orbits γ in that region for which the image of u intersects R × γ . This is aclosed subset of the connected topological space of all Reeb orbits in [1 / , N ] × T :indeed, if γ k ∈ Υ is a sequence converging to some orbit γ ∞ , then u ( z k ) ∈ R × γ k for some sequence z k ∈ R × S , which must be contained in a compact subset sincethe asymptotic orbits of u lie outside of [1 / , N ] × T , hence z k has a convergentsubsequence z k → z ∞ ∈ R × S with u ( z ∞ ) ∈ R × γ ∞ , proving γ ∞ ∈ Υ. We claimthat Υ is also an open subset of the space of orbits in [1 / , N ] × T . This follows Chris Wendl from positivity of intersections, as every R × γ is also a J -holomorphic curve: if u ( z ) ∈ R × γ , then for every other closed orbit γ ′ close enough to γ , there is a point z ′ ∈ R × S near z with u ( z ′ ) ∈ R × γ ′ . This proves that, in fact, u passes through R × γ for every orbit γ in the region [1 / , N ] × T . We will now use this to showthat if N is sufficiently large, the contact area of u will be larger than is allowed byStokes’ theorem.Let us write u ( s, t ) = ( r ( s, t ) , ρ ( s, t ) , φ ( s, t ) , θ ( s, t )) ∈ R × (cid:0) R (cid:14) (2 N + 2) Z (cid:1) × S × S and choose two points ρ ∈ [1 / ,
1] and ρ ∈ [ N − , N ] which are both regularvalues of the function ρ : R × S → R / (2 N + 2) Z . The intersections of u with theorbits in [1 / , N ] × T imply that the function ρ ( s, t ) attains every value in [1 / , N ],and since the asymptotic limits of u lie outside this region, U := ρ − ([ ρ , ρ ]) ⊂ R × S is then a nonempty and compact smooth submanifold with boundary ∂ U = − C ⊔ C , where C i := ρ − ( ρ i ) for i = 1 ,
2. Restricting u to the multicurves C i then gives apair of smooth maps w i : C i → T : ( s, t ) ( φ ( s, t ) , θ ( s, t )) , i = 1 , , which are homologous to each other. Denote the generators of H ( T ) correspondingto the φ - and θ -coordinates by ℓ φ and ℓ θ respectively, and suppose [ w i ] = mℓ φ + nℓ θ for m, n ∈ Z . The key observation now is that the restriction of α to each of thetori { ρ i } × T is a closed 1-form, thus for each i = 1 , R C i u ∗ α depends only onthe homology class mℓ φ + nℓ θ ∈ H ( T ) and not any further on the maps w i . Inparticular, Z C i u ∗ α = f ( ρ i ) n + g ( ρ i ) m for i = 1 ,
2. We now compute, Z U u ∗ dα = Z C u ∗ α − Z C u ∗ α = n [ f ( ρ ) − f ( ρ )] + m [ g ( ρ ) − g ( ρ )]= n [(1 − ǫδρ ) − (1 − ǫδρ )] + m [ ǫρ − ǫρ ]= ǫ ( ρ − ρ )( m − nδ )This integral has to be positive since u ∗ dα ≥ u is not a trivial cylinder, thus m − nδ >
0. Moreover, δ was assumed rational, so if δ = p/q for some p, q ∈ N , wehave m − nδ = 1 q ( mq − np ) ≥ q , implying Z R × S u ∗ dα ≥ Z U u ∗ dα ≥ ǫq ( ρ − ρ ) ≥ ǫ ( N − q . Having chosen δ (which determines q ) and ǫ in advance, we are free to make N aslarge as we like. But by (10.2), R R × S u ∗ dα cannot be any larger than the period ectures on Symplectic Field Theory of its positive asymptotic orbit, which does not depend on N . So this gives acontradiction, proving that u cannot touch the region { ρ ≥ / } . The mirror imageof this argument shows that u also cannot touch the region { ρ ≤ − / } . (cid:3) With Lemma 10.34 in hand, the calculation of HC h ∗ ( T N , α, J ) for sufficientlylarge N is straightforward: there is one odd generator and one even generator, witha trivial differential, giving HC h ∗ ( T , ξ ) ∼ = ( Z ∗ = odd , Z ∗ = even . This calculation can now be extended to ( T , ξ k ) by a cheap trick: using the contac-tomorphism ( T N , ker α ) → ( T , ξ ), let us identify T N with T and write α = F α for some function F : T → (0 , ∞ ). Then the k -fold covering mapΦ k : T → T : ( ρ, φ, θ ) ( kρ, φ, θ )maps the homotopy class h to itself and pulls back ξ to ξ k , so Φ ∗ k α is a contact formfor ξ k . It is also h -admissible: indeed, Φ ∗ k α admits no contractible orbits since theywould project down to contractible orbits on ( T , α ), and every orbit in P h (Φ ∗ k α )projects to one in P h ( α ), hence they are all nondegenerate. The almost complexstructure Φ ∗ k J ∈ J (Φ ∗ k α ) then makes the map Id × Φ k : ( R × T , Φ ∗ k J ) → ( R × T , J )holomorphic, so every Φ ∗ k J -holomorphic cylinder counted by HC h ∗ ( T , Φ ∗ k α, Φ ∗ k J )projects to a J -holomorphic cylinder counted by HC h ∗ ( T , α, J ), and conversely,each orbit in P h ( α ) and each J -holomorphic cylinder has exactly k lifts to the cover.The generators of CC h ∗ ( T , Φ ∗ k α ) thus consist of 2 k orbits, k odd and k even, with2 k connecting Φ ∗ k J -holomorphic cylinders that cancel each other in pairs, giving atrivial differential. In summary: HC h ∗ ( T , ξ k ) ∼ = ( Z k ∗ = odd , Z k ∗ = even . ECTURE 11
Coherent orientationsContents
This lecture will be concerned with orienting the moduli spaces M ( J ) := M g,m ( J, A, γ + , γ − )of J -holomorphic curves in a completed symplectic cobordism c W , in cases wherethey are smooth. We assume as usual that all Reeb orbits are nondegenerate sothat the usual linearized Cauchy-Riemann operators are Fredholm.For SFT and other Floer-type theories, it is not enough to know that eachcomponent of M ( J ) is orientable—relations like ∂ = 0 rely on having certaincompatibility conditions between the orientations on different components. Thepoint is that whenever a space of broken curves is meant to be interpreted as theboundary of some other compactified moduli space, we need to make sure that itcarries the boundary orientation. This compatibility is what is known as coherence ,and in order to define it properly, we need to return to the subject of gluing.Our discussion of gluing in Lecture 10 was fairly simple because it was limitedto somewhere injective holomorphic cylinders that could only break along simplycovered Reeb orbits. Recall however that more general holomorphic buildings carrya certain amount of extra structure that was not relevant in that simple case. Evenin a building u that has only two nontrivial levels u − and u + , the breaking puncturescarry decorations : i.e. if { z + , z − } is a breaking pair in u , then the decoration definesan orientation-reversing orthogonal map δ z + Φ −→ δ z − between the two “circles at infinity” δ z ± associated to the punctures z ± (see § Chris Wendl cannot be deduced from knowledge of u − and u + alone. We therefore need toconsider moduli spaces of curves with a bit of extra structure.For each Reeb orbit γ in M + or M − , choose a point on its image p γ ∈ im γ ⊂ M ± . For a J -holomorphic curve u : ( ˙Σ = Σ \ (Γ + ∪ Γ − ) , j ) → ( c W , J ) with a puncture z ∈ Γ ± asymptotic to γ , an asymptotic marker is a choice of a ray ℓ ⊂ T z Σ suchthat lim t → + u ( c ( t )) = ( ±∞ , p γ )for any smooth path c ( t ) ∈ Σ with c (0) = z and 0 = ˙ c (0) ∈ ℓ . If γ has coveringmultiplicity m ∈ N , then there are exactly m choices of asymptotic markers at z ,related to each other by the action on T z Σ by the m th roots of unity. We shalldenote M $ ( J ) := M $ g,m ( J, A, γ + , γ − ) := (cid:8) (Σ , j, Γ + , Γ − , Θ , u, ℓ ) (cid:9) (cid:14) ∼ , where (Σ , j, Γ + , Γ − , Θ , u ) represents an element of M g,m ( J, A, γ + , γ − ), ℓ denotes anassignment of asymptotic markers to every puncture z ∈ Γ ± , and(Σ , j , Γ +0 , Γ − , Θ , u , ℓ ) ∼ (Σ , j , Γ +1 , Γ − , Θ , u , ℓ )means the existence of a biholomorphic map ψ : (Σ , j ) → (Σ , j ) which definesan equivalence of (Σ , j , Γ +0 , Γ − , Θ , u ) with (Σ , j , Γ +1 , Γ − , Θ , u ) and satisfies ψ ∗ ℓ = ℓ . There is a natural surjection M $ ( J ) → M ( J )defined by forgetting the markers. We will say that an element u ∈ M $ ( J ) isFredholm regular whenever its image under the map to M ( J ) is regular. Let M $ , reg ( J ) = M $ , reg g,m ( J, A, γ + , γ − ) ⊂ M $ ( J )denote the open subset consisting of Fredholm regular curves with asymptotic mark-ers. Note that components of M ( J ) and M $ ( J ) consisting of closed curves areidentical spaces; components with punctures have the following simple relationshipto each other. Proposition . Each component of M $ , reg ( J ) consisting of curves with atleast one puncture admits the structure of a smooth manifold, whose dimension oneach connected component matches that of M reg ( J ) . Moreover, the natural map M $ , reg ( J ) → M reg ( J ) is smooth, and the preimage of a curve u ∈ M reg ( J ) with asymptotic orbits { γ z } z ∈ Γ of covering multiplicities { κ z } z ∈ Γ contains exactly Q z ∈ Γ κ z | Aut( u ) | distinct elements. ectures on Symplectic Field Theory Proof.
The smooth structure of M $ , reg ( J ) arises from the same argument weused in Lecture 7 for M reg ( J ), supplemented by the following remarks: first, ev-ery nontrivial automorphism ψ ∈ Aut( u ) for u ∈ M ( J ) acts nontrivially on theasymptotic markers. Indeed, ψ is required to fix each of the punctures and is a bi-holomorphic map with ψ k ≡ Id for some k ∈ N , thus it takes the form z e πim/k insuitable holomorphic coordinates near each puncture for suitable integers m, k ∈ Z .If m = 0, then unique continuation implies ψ ≡ Id, and otherwise ψ changes theasymptotic marker at every puncture. With this understood, one can define asin § M $ ( J ) with ¯ ∂ − J (0) / Aut(Σ , j , Γ ∪ Θ), where ¯ ∂ − J (0)includes information about asymptotic markers and is a smooth manifold by the im-plicit function theorem, but Aut(Σ , j , Γ ∪ Θ) acts on it freely , producing a quotientwith no isotropy.Finally, if (Σ , j, Γ ∪ Θ , u ) represents an element of M ( J ) with asymptotic orbits { γ z } z ∈ Γ , then the number of possible choices of asymptotic markers is precisely Q z ∈ Γ κ z . However, not all of these produce inequivalent elements of M $ ( J ): indeed,the previous paragraph shows that Aut( u ) acts freely on the set of all choices ofmarkers, so that the total number of inequivalent choices is as stated. (cid:3) Suppose u + and u − are two (possibly disconnected and/or nodal) holomorphiccurves, with asymptotic markers, such that the number of negative punctures of u + equals the number of positive punctures of u − , and the asymptotic orbit of u + atits i th negative puncture matches that of u − at its i th positive puncture for every i .Then the pair ( u − , u + ) naturally determines a holomorphic building: indeed, thebreaking punctures admit unique decorations determined by identifying the markerson u + with the markers at corresponding punctures of u − .Let us now consider a concrete example of a gluing scenario. Figure 11.1 showsthe degeneration of a sequence of curves in M , ( J, A k , ( γ , γ ) , γ − ) to a building u ∈ M , ( J, A + B + C, ( γ , γ ) , γ − ) with one main level and one upper level. Themain level is a connected curve u A ∈ M , ( J, A, ( γ , γ , γ ) , γ − ), and the upper levelconsists of two connected curves u B ∈ M , ( J + , B, γ , ( γ , γ )) , u C ∈ M , ( J + , C, γ , γ ) . One can endow each of these curves with asymptotic markers compatible with thedecoration of u ; this is a non-unique choice, but e.g. if one chooses markers for u A arbitrarily, then the markers at the negative punctures of u B and u C are uniquelydetermined. Now if all three curves are Fredholm regular, then a substantial general-ization of the gluing procedure outlined in Lecture 10 provides open neighborhoods U $ A and U $ BC , u A ∈ U $ A ⊂ M $1 , ( J, A, ( γ , γ , γ )) , [( u B , u c )] ∈ U $ BC ⊂ (cid:0) M $1 , ( J + , B, γ , ( γ , γ )) × M $0 , ( J + , C, γ , γ ) (cid:1) . R which are smooth manifolds of dimensionsdim U $ A = vir-dim M , ( J, A, ( γ , γ , γ )) , dim U $ BC = vir-dim M , ( J + , B, γ , ( γ , γ )) + vir-dim M , ( J + , C, γ , γ ) − , Chris Wendl u A u B u C γ − γ − γ γ γ γ γ γ γ c W c W R × M + Figure 11.1.
The degeneration scenario behind the gluing map (11.1)along with a smooth embedding(11.1) Ψ : [ R , ∞ ) × U $ A × U $ BC ֒ → M $3 , ( J, A + B + C, ( γ , γ ) , γ − ) , defined for R ≫
1. This is an example of a gluing map : it has the property thatfor any u ∈ U $ A and v ∈ U $ BC , Ψ( R, u, v ) converges in the SFT topology as R → ∞ to the unique building (with asymptotic markers) having main level u and upperlevel v , and moreover, every sequence of smooth curves degenerating in this way iseventually in the image of Ψ.In analogous ways one can define gluing maps for buildings with a main level anda lower level, or more than two levels, or multiple levels in a symplectization (alwaysdividing symplectization levels by the R -action). It’s important to notice that in allsuch scenarios, the domain and target of the gluing map have the same dimension,e.g. the dimension of both sides of (11.1) is the sum of the virtual dimensions of thethree moduli spaces concerned. Definition . A set of orientations for the connected components of M $ ( J )and M $ ( J ± ) is called coherent if all gluing maps are orientation preserving.Stated in this way, this definition is based on the pretense that we never have toworry about non-regular curves in any components of M $ ( J ), and that is of coursefalse—sometimes regularity cannot be achieved, in particular for multiply coveredcurves. As we’ll see though in § ectures on Symplectic Field Theory of regularity (see Definition 11.14). The main result whose proof we will outline inthe next few sections is then: Theorem . Coherent orientations exist.
But there is also some bad news. The space M $ ( J ) with asymptotic markersis not actually the space we want to orient. In fact, even the usual moduli space M ( J ) has a certain amount of extra information in it that we’d rather not keeptrack of when we don’t have to, for instance the ordering of the punctures. Can weforget this information without forgetting the orientation of the moduli space? Notalways: Proposition . Suppose ˆ γ + = ( γ +1 , . . . , γ + k + ) , and ˇ γ + is a similar ordered listof Reeb orbits obtained from ˆ γ + by exchanging γ + j with γ + k for some ≤ j < k ≤ k + .Then for any choice of coherent orientations, the natural map M $ g,m ( J, A, ˆ γ + , γ − ) → M $ g,m ( J, A, ˇ γ + , γ − ) defined by permuting the corresponding punctures z + j , z + k ∈ Γ + along with their as-ymptotic markers is orientation reversing if and only if the numbers n − µ CZ ( γ + i ) for i = j, k are both odd. A similar statement holds for permutations of negativepunctures. This result is the reason for the super-commutative algebra that we will see inthe next lecture. What about forgetting the markers? It turns out that we cansometimes do that as well, but again not always.
Proposition . Suppose M $ g,m ( J, A, γ + , γ − ) → M $ g,m ( J, A, γ + , γ − ) is themap defined by multiplying the asymptotic marker by e πi/m at one of the puncturesfor which the asymptotic orbit is an m -fold cover γ m of a simple orbit γ . For anychoice of coherent orientations, this map reverse orientation if and only if m is evenand µ CZ ( γ m ) − µ CZ ( γ ) is odd. Note that in both of the above propositions, only the odd/even parity of theConley-Zehnder indices matters, so there is no need to choose trivializations. Propo-sition 11.5 motivates one of the more mysterious technical definitions in SFT.
Definition . A closed nondegenerate Reeb orbit γ is called a bad orbit ifit is an m -fold cover of some simple orbit γ ′ where m is even and µ CZ ( γ ) − µ CZ ( γ ′ )is odd. Orbits that are not bad are called good .The upshot is that coherent orientations can be defined on the union of allcomponents M g,m ( J, A, γ + , γ − ) for which all of the orbits in the lists γ + and γ − aregood. This does not mean that moduli spaces involving bad orbits cannot be dealtwith—in fact, such moduli spaces have the convenient property that the number ofdistinct choices of asymptotic markers is always even, and every such choice can becancelled by an alternative choice that induces the opposite orientation. For thisreason, while bad orbits certainly can appear in breaking of holomorphic curves, wewill see that they do not need to serve as generators of SFT. Chris Wendl
Before addressing the actual construction of coherent orientations, we can al-ready give heuristic proofs of Propositions 11.4 and 11.5. They are not fully rigor-ous because they are based on the same pretense as Definition 11.2, namely thatall curves we ever have to worry about (including multiple covers) are regular. Butwe will be able to turn these into precise arguments in § Heuristic proof of Proposition 11.4.
To simplify the notation, supposeˆ γ + consists of only two orbits, so ˆ γ + = ( γ , γ ) and ˇ γ + = ( γ , γ ). Consider thegluing scenario shown in Figure 11.2, where u ∈ M $ g,m ( J, A, ( γ , γ ) , γ − ) needs tobe glued to a disjoint union of two planes u B ∈ M $0 , ( J + , B, ∅ , γ ) , u C ∈ M $0 , ( J + , C, ∅ , γ ) . You might object that there’s no guarantee that such planes must exist in R × M + ,e.g. the orbits γ and γ might not even be contractible. This concern is valid sofar as it goes, but it misses the point: since we’re talking about gluing rather thancompactness, we do not need any seriously global information about c W and M + ,as the gluing process doesn’t depend on anything outside a small neighborhood ofthe curves we’re considering. Thus we are free to change the global structure of M + elsewhere so that the planes u B and u C will exist. If you still can’t imaginehow one might do this, try not to worry about it and just think of Figure 11.2 as athought-experiment: it’s a situation that certainly does sometimes happen, so whenit does, let’s see what it implies about orientations.Assuming all three curves in the picture are regular, there will be smooth openneighborhoods u ∈ U ⊂ M $ g,m ( J, A, ( γ , γ ) , γ − )[( u B , u C )] ∈ U BC ⊂ (cid:0) M $0 , ( J + , B, ∅ , γ ) × M $0 , ( J + , C, ∅ , γ ) (cid:1) . R and a gluing mapΨ BC : [ R , ∞ ) × U × U BC ֒ → M $ g,m ( J, A + B + C, ∅ , γ − ) , which must be orientation preserving by assumption. But reversing the order of theproduct M $0 , ( J + , B, ∅ , γ ) ×M $0 , ( J + , C, ∅ , γ ) and letting u ′ ∈ M $ g,m ( J, A, ( γ , γ ) , γ − )denote the image of u under the map that switches the order of its positive punc-tures, there are also smooth open neighborhoods u ′ ∈ U ⊂ M $ g,m ( J, A, ( γ , γ ) , γ − )[( u C , u B )] ∈ U CB ⊂ (cid:0) M $0 , ( J + , C, ∅ , γ ) × M $0 , ( J + , B, ∅ , γ ) (cid:1) . R and a gluing mapΨ CB : [ R , ∞ ) × U × U CB ֒ → M $ g,m ( J, A + B + C, ∅ , γ − ) . Of course by the maximum principle, planes with only negative ends will not exist in R × M + if this is the symplectization of a contact manifold. But we could also change the contact data toa stable Hamiltonian structure for which such planes are allowed. ectures on Symplectic Field Theory u B u C u γ − γ − γ γ c W c W R × M + Figure 11.2.
The gluing thought-experiment used for provingPropositions 11.4 and 11.5.If both of these gluing maps preserve orientation, then the effect on orientations ofthe map from M $ g,m ( J, A, ( γ , γ ) , γ − ) to M $ g,m ( J, A, ( γ , γ ) , γ − ) defined by inter-changing the positive punctures must be the same as that of the map M $0 , ( J + , B, ∅ , γ ) × M $0 , ( J + , C, ∅ , γ ) → M $0 , ( J + , C, ∅ , γ ) × M $0 , ( J + , B, ∅ , γ )( u B , u C ) ( u C , u B ) . The latter is orientation reversing if and only if both moduli spaces of planes areodd dimensional, which means n − µ CZ ( γ i ) is odd for i = 1 , (cid:3) Heuristic proof of Proposition 11.5.
Let us reuse the thought-experimentof Figure 11.2, but with different details in focus. Suppose γ in the picture is an m -fold covered orbit γ m , where γ is simply covered, and suppose that u B is also an m -fold cover, taking the form u B ( z ) = v ( z m )for a somewhere injective plane v ∈ M , ( J + , B , ∅ , γ ). We’re going to assume againthat all curves in the discussion are regular, including the multiple cover u B ; whilethis doesn’t sound very plausible, we will see once the determinant line bundle entersthe picture in § u B has a cyclic automorphismgroup Aut( u B ) = Z m ⊂ U(1)which acts freely on the set of m choices of asymptotic marker for u B . Then if weact with the same element of Z m on u B and on the corresponding asymptotic marker Chris Wendl for u , the building is unchanged, as it has the same decoration. Coherence thereforeimplies that the effect on orientations of the map from M $ g,m ( J, A, ( γ , γ ) , γ − ) toitself defined by acting with the canonical generator of Z m ⊂ U(1) on the marker at γ is the same as the effect of the map M $0 , ( J + , mB , ∅ , γ m ) → M $0 , ( J + , mB , ∅ , γ m )defined by composing u B : C → R × M + with ψ ( z ) := e πi/m z .The derivative of this map from M $0 , ( J + , mB , ∅ , γ m ) to itself at u B defines alinear self-mapΨ : T u B M , ( J + , mB , ∅ , γ m ) → T u B M , ( J + , mB , ∅ , γ m )with Ψ m = . The latter implies that Ψ cannot reverse orientation if m is odd. If m is even, observe that the representation theory of Z m gives a decomposition T u B M , ( J + , mB , ∅ , γ m ) = V ⊕ V − ⊕ V rot , where Ψ acts on V ± as ± , and V rot is a direct sum of real 2-dimensional subspaceson which Ψ acts by rotations (and therefore preserves orientations). Thus Ψ reversesthe orientation of T u B M , ( J + , mB , ∅ , γ m ) if and only if dim V − is odd. As we willreview in the next section, T u B M , ( J + , mB , ∅ , γ m ) is a space of holomorphic sec-tions of u ∗ B T ( R × M + ) modulo a subspace defined via the linearized automorphismsof C , so V consists of precisely those sections η that satisfy η = η ◦ ψ , meaning theyare m -fold covers of sections of v ∗ T ( R × M + ). This defines a bijective correspondencebetween V and T v M , ( J + , B , ∅ , γ ), sodim V − = dim M , ( J + , mB , ∅ , γ m ) − dim M , ( J + , B , ∅ , γ ) (mod 2) . The result then comes from plugging in the dimension formulas for these two modulispaces. (cid:3)
We now discuss concretely what is involved in orienting a moduli space of J -holomorphic curves.Recall from Lecture 7 that whenever a curve u : ( ˙Σ = Σ \ Γ , j ) → ( c W , J ) withmarked points Θ ⊂ ˙Σ is Fredholm regular, a neighborhood of u in M ( J ) can beidentified with ¯ ∂ − J (0) (cid:14) G , where G = Aut(Σ , j , Γ ∪ Θ) and ¯ ∂ J is the smooth Fredholm section T × B k,p,δ → E k − ,p,δ : ( j, u ) T u + J ◦ T u ◦ j, defined on the product of a G -invariant Teichm¨uller slice T through j with aBanach manifold B k,p,δ of W k,p -smooth maps ˙Σ → c W satisfying an exponentialdecay condition at the cylindrical ends. Here G acts on ¯ ∂ − J (0) by(11.2) G × ¯ ∂ − J (0) → ¯ ∂ − J (0) : ( ϕ, ( j, u )) ( ϕ ∗ j, u ◦ ϕ ) . Regularity means that the linearization D ¯ ∂ J ( j , u ) : T j T ⊕ T u B k,p,δ → E k − ,p,δ ( j ,u ) issurjective, and the implicit function theorem then gives a natural identification T u M ( J ) = ker D ¯ ∂ J ( j , u ) (cid:14) aut (Σ , j , Γ ∪ Θ) , ectures on Symplectic Field Theory where aut (Σ , j , Γ ∪ Θ) denotes the Lie algebra of G , which acts on ker D ¯ ∂ J ( j , u ) bydifferentiating (11.2). This action actually defines an inclusion of aut (Σ , j , Γ ∪ Θ)into ker D ¯ ∂ J ( j , u ) whenever u is not constant, thus we can regard aut (Σ , j , Γ ∪ Θ)as a subspace of ker D ¯ ∂ J ( j , u ).As outlined in Proposition 11.1, the space M $ ( J ) with asymptotic markers ad-mits a similar local description: here one only needs to enhance the structure of theBanach manifold B k,p,δ with information about asymptotic markers at each punc-ture, so the Banach manifold needed to describe M $ ( J ) is a finite covering spaceof B k,p,δ . The rest of the discussion is identical, except for the fact that when markersare included, G always acts freely on ¯ ∂ − J (0).We now make a useful observation about the spaces aut (Σ , j , Γ ∪ Θ) and T j T :namely, they both carry natural complex structures and are thus canonically ori-ented. This follows from the fact that both the automorphism group G and theTeichm¨uller space T (Σ , Γ ∪ Θ) = J (Σ) (cid:14) Diff (Σ , Γ ∪ Θ) are naturally complex mani-folds. On the linearized level, one way to see it is via the fact—mentioned previouslyin § aut (Σ , j , Γ ∪ Θ) and T [ j ] T (Σ , Γ ∪ Θ) can be naturally identified withthe kernel and cokernel respectively of the natural linear Cauchy-Riemann typeoperator on (Σ , j ),(11.3) D Id : W k,p Γ ∪ Θ ( T Σ) → W k − ,p (End C ( T Σ)) , which is the linearization at Id of the nonlinear operator that detects holomorphicmaps (Σ , j ) → (Σ , j ). This operator is equivalent to the operator that defines theholomorphic structure of T Σ, thus it is complex linear. To handle the puncturesand marked points, one needs to restrict the nonlinear operator to the space of W k,p -smooth maps Σ → Σ that fix every point in Γ ∪ Θ, thus the domain of thelinearization becomes the finite-codimensional subspace W k,p Γ ∪ Θ ( T Σ) := (cid:8) X ∈ W k,p ( T Σ) (cid:12)(cid:12) X | Γ ∪ Θ = 0 (cid:9) . This subspace is still complex, thus so is (11.3), and its kernel and cokernel inheritnatural complex structures.The complex structure on aut (Σ , j , Γ ∪ Θ) means that defining an orientation onthe tangent space T u M $ ( J ) is equivalent to defining one on ker D ¯ ∂ J ( j , u ). Thelatter operator takes the form D ¯ ∂ J ( j , u ) : T j T ⊕ T u B k,p,δ → E k − ,p,δ ( j ,u ) : ( y, η ) J ◦ T u ◦ y + D u η, where D u : W k,p,δ ( u ∗ T c W ) ⊕ V Γ → W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )) is the usual lin-earized Cauchy-Riemann operator at u , with V Γ denoting a complex ( u is J -holomorphic imply that the first term in this operator, T j T → E k − ,p,δ ( j ,u ) : y J ◦ T u ◦ y The presence of aut (Σ , j , Γ ∪ Θ) in this discussion is only relevant in the finite set of “non-stable” cases where χ ( ˙Σ \ Θ) ≥
0, since otherwise G is finite and thus aut (Σ , j , Γ ∪ Θ) is trivial. Chris Wendl is a complex-linear map. Now if D u happens also to be a complex-linear map, thenwe are done, because ker D ¯ ∂ J ( j , u ) will then be a complex vector space and inherita natural orientation.In general, D u is not complex linear, though it does have a complex-linear part , D C u η := 12 ( D u η − J D u ( J η )) , which is also a Cauchy-Riemann type operator. The space of all Cauchy-Riemanntype operators on a fixed vector bundle is affine, so one can interpolate from D u to D C u through a path of Cauchy-Riemann type operators, though they may not allbe Fredholm—this depends on the asymptotic operators at the punctures. In thespecial case however where there are no punctures, one can easily imagine makinguse of this idea: if ˙Σ = Σ is a closed surface, then the obvious homotopy from D u to its complex-linear part yields a homotopy from D ¯ ∂ J ( j , u ) to its complex-linearpart, and if every operator along this homotopy happens to be surjective, then thecanonical orientation defined on the kernel of the complex-linear operator determinesan orientation on ker D ¯ ∂ J ( j , u ).There are two obvious problems with the above discussion:(1) We have no way to ensure that every operator in the homotopy from D ¯ ∂ J ( j , u ) to its complex-linear part is surjective;(2) If there are punctures, then we cannot even expect every operator in thishomotopy to be Fredholm.The first problem motivates the desire to define a notion of orientations for aFredholm operator T that does not require T to be surjective but reduces to theusual notion of orienting ker T whenever it is. The solution to this problem isthe determinant line bundle , which we will discuss in the next section. With thisobject in hand, the above discussion for the case of closed curves can be maderigorous, so that all smooth moduli spaces of closed J -holomorphic curves inheritcanonical orientations. One of the advantages of using the determinant line bundleis that the question of orientations becomes entirely disjoined from the question oftransversality: if one can orient the determinant line bundle then moduli spaces ofregular curves inherit orientations, but orienting the determinant bundle does notrequire knowing in advance whether the curves are regular.The second problem is obviously significant because in the punctured case, mod-uli spaces of J -holomorphic curves sometimes have odd real dimension, making itclearly impossible to homotop D ¯ ∂ J ( j , u ) through Fredholm operators to one thatis complex linear. The solution in this case will be to define orientations algorithmi-cally via the coherence condition, and we will describe a suitable algorithm for thisin § Fix real Banach spaces X and Y and let Fred R ( X, Y ) denote the space of real-linear Fredholm operators, viewed as an open subset of the Banach space L R ( X, Y )of all bounded linear operators. We’ll use the following notation throughout: if V ectures on Symplectic Field Theory is an n -dimensional real vector space, then the top-dimensional exterior power of V is denoted by Λ max V := Λ n V. This 1-dimensional real vector space is spanned by any wedge product of the form v ∧ . . . ∧ v n where ( v , . . . , v n ) is a basis of V . Denoting the dual space of V by V ∗ ,note that there is a canonical isomorphism (Λ max V ) ∗ = Λ max V ∗ . If dim V = 0, thenwe adopt the convention Λ max V = R . Definition . Given T ∈ Fred R ( X, Y ), the determinant line of T is thereal 1-dimensional vector spacedet( T ) = (Λ max ker T ) ⊗ (Λ max coker T ) ∗ . Our main goal in this section is to prove:
Theorem . There exists a topological vector bundle det(
X, Y ) π −→ Fred R ( X, Y ) of real rank such that π − ( T ) = det( T ) for each T ∈ Fred R ( X, Y ) . Observe that whenever T ∈ Fred R ( X, Y ) is surjective, det( T ) = Λ max ker T , soan orientation of det( T ) is equivalent to an orientation of ker T . More generally,an orientation of det( T ) is equivalent to an orientation for ker T ⊕ coker T . If T isan isomorphism, then det( T ) is simply R , so an orientation of det( T ) amounts to achoice of sign ± X, Y ) → Fred R ( X, Y ), we start withthe case where X and Y are both finite dimensional. Note that in this case, everylinear map is Fredholm, including the zero map, and its determinant is simplyΛ max X ⊗ (Λ max Y ) ∗ . Lemma . Suppose X and Y are real vector spaces of finite dimensions n and m respectively. Then for every T ∈ L R ( X, Y ) , there exists a canonical isomorphism (Λ max ker T ) ⊗ (Λ max coker T ) ∗ = (Λ max V ) ⊗ (Λ max W ) ∗ . Proof.
Suppose dim ker T = k and dim coker T = ℓ , so ind( T ) = k − ℓ = n − m ,thus n − k = m − ℓ . We define a linear map Φ : (Λ n X ) ⊗ (Λ m Y ) ∗ → (cid:0) Λ k ker T (cid:1) ⊗ (cid:0) Λ ℓ coker T (cid:1) ∗ via the following procedure. Fix x ∈ Λ n X and y ∗ ∈ (Λ m Y ) ∗ andsuppose both are nontrivial. Then for any nontrivial element k ∈ Λ k ker T , thereexists a unique element v ∈ Λ n − k ( X/ ker T ) such that for any subspace V ⊂ X complementary to ker T , the element ˜ v ∈ Λ n − k V ⊂ Λ n − k X obtained from v byinverting the natural isomorphism V → X/ ker T induced by the projection X → X/ ker T satisfies k ∧ ˜ v = x . The map T descends to an isomorphism X/ ker T → im T and thus induces an iso-morphism Λ n − k ( X/ ker T ) → Λ m − ℓ (im T ) ⊂ Λ m − ℓ Y , which takes v to a nontrivialelement Tv . There is then a unique element c ∈ Λ ℓ coker T = Λ ℓ ( Y / im T ) suchthat for any subspace W ⊂ Y complementary to im T , the element ˜ c ∈ Λ ℓ W ⊂ Λ ℓ Y obtained from c by inverting the isomorphism W → Y / im T induced by the pro-jection Y → Y / im T satisfies y ∗ (˜ c ∧ Tv ) = 1 . Chris Wendl
Now define Φ as the unique linear map such thatΦ( x ⊗ y ∗ ) = k ⊗ c ∗ , where c ∗ ∈ (Λ ℓ coker T ) ∗ is defined by c ∗ ( c ) = 1. It is straightforward to checkthat this definition does not depend on any choices: indeed, if we replace k by λ k for some λ ∈ R \ { } in the above procedure, then v is replaced by λ v , hence Tv becomes λ Tv , c becomes λ c and c ∗ therefore becomes λ c ∗ , so that k ⊗ c ∗ is replacedby ( λ k ) ⊗ (cid:18) λ c ∗ (cid:19) = k ⊗ c ∗ . (cid:3) To construct local trivializations of det(
X, Y ) in the infinite-dimensional case,recall the following construction from Lecture 3. Given T ∈ Fred R ( X, Y ), we canwrite X = V ⊕ K and Y = W ⊕ C where K = ker T , C ∼ = coker T , W = im T and T | V : V → W is an isomorphism. We shall use these splittings to write anyother operator T ∈ Fred R ( X, Y ) as T = (cid:18) A BC D (cid:19) and let
U ⊂
Fred R ( X, Y ) denote the open neighborhood of T for which the block A : V → W is invertible. This gives rise to a pair of smooth mapsΦ : U → L R ( K, C ) : T D − CA − B and F : U → L R ( V ⊕ K ) = L R ( X ) : T (cid:18) − A − B (cid:19) , such that F ( T ) is always invertible and maps { }⊕ ker Φ( T ) isomorphically to ker T .Similarly, there is a smooth map G : U → L R ( W ⊕ C ) = L R ( Y ) : T (cid:18) − CA − (cid:19) such that G ( T ) is always invertible and maps im T isomorphically to W ⊕ im Φ( T ),so it descends to an isomorphism of coker T to coker Φ( T ). Given the canonical iso-morphism det(Φ( T )) = Λ max K ⊗ (Λ max C ) ∗ = det( T ) from Lemma 11.9, the result-ing smooth families of isomorphisms ker T → ker Φ( T ) and coker T → coker Φ( T )determine a local trivializationdet( X, Y ) | U → U × det( T ) . I will leave it as an exercise for the reader to check that the resulting transitionmaps are continuous. This detail should not be underestimated, e.g. [ MW , § MS04 , § A.2] are, unfortunately, not continuously compatible. See [
Zin ] forfurther discussion of this point. If you discover that my local trivializations are also not continu-ously compatible, please let me know. ectures on Symplectic Field Theory
Exercise . Show that if X and Y are complex Banach spaces, thenthe restriction of det( X, Y ) to the subspace of complex-linear Fredholm operatorsFred C ( X, Y ) ⊂ Fred R ( X, Y ) admits a canonical orientation compatible with thecomplex structures of ker T and coker T for each T ∈ Fred C ( X, Y ). Show also thatwhenever T ∈ Fred C ( X, Y ) is an isomorphism, the canonical orientation of det( T )agrees with the standard orientation of R .The orientation of det( T ) for T ∈ Fred C ( X, Y ) described in Exercise 11.10 iscalled the complex orientation . Combining ideas from the previous two sections, letdet( J ) → M $ ( J )denote the topological line bundle that associates to any u ∈ M $ g,m ( J, A, γ + , γ − )the determinant line of the Fredholm operator D u : W k,p,δ ( u ∗ T c W ) ⊕ V Γ → W k − ,p,δ (Hom C ( T ˙Σ , u ∗ T c W )) . One can construct local trivializations for this bundle using Theorem 11.8 and anychoice of local trivializations for the Banach space bundles T B k,p,δ and E k − ,p,δ . Proposition . Any orientation of det( J ) → M $ ( J ) canonically deter-mines an orientation of M reg ( J ) . Proof.
As explained in § M reg ( J ) near a particular curve u : ( ˙Σ , j ) → ( c W , J ) is equivalent to a continuously varying choice of orientationsfor the kernels ker D ¯ ∂ J ( j, u ) ⊂ T j T ⊕ T u B k,p,δ for all ( j, u ) ∈ ¯ ∂ − J (0), where T is a Teichm¨uller slice through j . The operator D ¯ ∂ J ( j, u ) is of the form L ( y, η ) := J ◦ T u ◦ y + D u η and thus is homotopic through Fredholm operators to L ( y, η ) := D u η, namely via the homotopy L s ( y, η ) := sJ ◦ T u ◦ y + D u η for s ∈ [0 , L are T j T ⊕ ker D u and coker D u respectively, and since T j T carries acomplex structure, the orientation of det( D u ) naturally determines an orientationof det( L ). Using the homotopy L s , this determines orientations of det( D ¯ ∂ J ( j, u ))and thus orientations of ker D ¯ ∂ J ( j, u ) for all ( j, u ) near ( j , u ), and this orientationdoes not depend on the choice of Teichm¨uller slice since the operators D u also donot. (cid:3) From now on, when we speak of an orientation of M $ ( J ), we will actuallymean an orientation of the bundle det( J ) → M $ ( J ). The above proposition impliesthat this is equivalent to what we want in applications, but one advantage of talkingabout det( J ) is that there is no need to limit the discussion to curves that are Chris Wendl regular, i.e. the notion of an orientation of M $ ( J ) now makes sense even though M $ ( J ) is not globally a smooth object. Proposition . Suppose all Reeb orbits in γ ± have the property that theirasymptotic operators are complex linear. Then M $ g,m ( J, A, γ + , γ − ) admits a naturalorientation, known as the complex orientation . Proof.
Having complex-linear asymptotic operators implies that the obvioushomotopy from each Cauchy-Riemann operator D u to its complex-linear part doesnot change the asymptotic operators and is therefore a homotopy through Fred-holm operators. We therefore have a continuously varying homotopy of each of therelevant fibers of det( J ) to the determinant bundle over a family of complex-linearoperators, which inherit the complex orientation described in Exercise 11.10. (cid:3) Proposition 11.12 applies in particular to all moduli spaces of closed J -holomorphiccurves, and thus solves the orientation problem in that case. We now briefly describe the construction of coherent orientations due to Bour-geois and Mohnke [
BM04 ]. A slightly different construction is described in [
EGH00 ],though it appears to have minor errors in some details.Recall from Lecture 4 the notion of an asymptotically Hermitian vector bundle(
E, J ) over a punctured Riemann surface ( ˙Σ , j ). Here ( ˙Σ , j ) is endowed with theextra structure of fixed cylindrical ends ( ˙ U z , j ) ∼ = ( Z ± , i ) for each puncture z ∈ Γ ± ,which determines a choice of asymptotic markers. Likewise, the bundle E comeswith an asymptotic bundle ( E z , J z , ω z ) → S associated to each puncture, carryingcompatible complex and symplectic structures. We shall now endow E with a bitmore structure that is always naturally present in the case E = u ∗ T c W : namely,assume each of the asymptotic bundles comes with a splitting(11.4) ( E z , J z , ω z ) = ( C ⊕ b E z , i ⊕ ˆ J z , ω ⊕ ˆ ω z ) , where ω is the standard symplectic structure on the trivial complex line bundle( C , i ) over S , and ( b E z , ˆ J z , ˆ ω z ) → S is another Hermitian bundle. Fix a choice { A z } z ∈ Γ of nondegenerate asymptotic operators on each of the bundles ( b E z , ˆ J z , ˆ ω z ),and define the topological space C R ( E, { A z } z ∈ Γ )to consist of all Cauchy-Riemann type operators on E that are asymptotic at thepunctures z ∈ Γ to the asymptotic operators( − i∂ t ) ⊕ A z : Γ( C ⊕ b E z ) → Γ( C ⊕ b E z ) . This is an affine space, so it is contractible, and if δ > V Γ ⊂ Γ( E ) denotes a complex ( C ⊕ { } ⊂ E z near each puncture z , then every D ∈ C R ( E, { A z } z ∈ Γ ) determines a Fredholm operator D : W k,p,δ ( E ) ⊕ V Γ → W k − ,p,δ (Hom C ( T Σ , E )) . ectures on Symplectic Field Theory It follows that a choice of orientation of the determinant line for any one of theseoperators determines an orientation for all of them. The point of this constructionis that every u ∈ M $ ( J ) determines an operator D u belonging to a space of thisform.We now construct a gluing operation for Cauchy-Riemann operators that lin-earizes the gluing maps described in § E i , J i ) → ( ˙Σ i = Σ i \ Γ i , j i ) for i = 0 , { A z } z ∈ Γ i , and thatthere exists a pair of punctures z ∈ Γ +0 and z ∈ Γ − such that some unitary bundleisomorphism b E z ∼ = −→ b E z identifies A z with A z . Note that such an isomorphism is uniquely determined upto homotopy whenever it exists. For R >
0, we can define a family of glued Riemannsurfaces ( ˙Σ R = Σ R \ Γ R , j R )by cutting off the ends ( R, ∞ ) × S ⊂ ˙ U z and ( −∞ , − R ) × S ⊂ ˙ U z and gluing { R } × S ⊂ ˙Σ to {− R } × S ⊂ ˙Σ . The glued Riemann surface contains anannulus biholomorphic to ([ − R, R ] × S , i ) in place of the infinite cylindrical endsat the punctures z and z . The unitary isomorphism b E z → b E z then determinesan isomorphism E z → E z via the splitting (11.4) and hence an asymptoticallyHermitian bundle ( E R , J R ) → ( ˙Σ R , J R ) . Using cutoff functions in the neck [ − R, R ] × S , any Cauchy-Riemann operators D i ∈ C R ( E i , { A z } z ∈ Γ i ) for i = 0 , D R ∈ C R ( E R , { A z } z ∈ Γ R )uniquely up to homotopy. Analogously to the gluing maps in § D R converge in some sense to the pair( D , D ) as R → ∞ , which has the following consequence: Lemma
BM04 , Corollary 7]) . For
R > sufficiently large, there is anatural isomorphism det( D ) ⊗ det( D ) → det( D R ) that is defined up to homotopy. (cid:3) Up to some additional direct sums and quotients by finite-dimensional complexvector spaces, this isomorphism should be understood as the linearization of a glu-ing map between moduli spaces, generalized to a setting in which the holomorphiccurves involved need not be regular. To orient M $ ( J ) coherently, it now suffices tochoose orientations for the operators in C R ( E, { A z } z ∈ Γ ) that vary continuously un-der deformations of j and E and are preserved by the isomorphisms of Lemma 11.13.This motivates the following generalization of Definition 11.2. Definition . A system of coherent orientations is an assignment toeach asymptotically Hermitian bundle (
E, J ) → ( ˙Σ , j ) with asymptotic splittings asin (11.4) and asymptotic operators { A z } z ∈ Γ of an orientation for the determinant Chris Wendl line of each D ∈ C R ( E, { A z } ), such that these orientations vary continuously with D as well as the data j and J , and such that the isomorphisms in Lemma 11.13 arealways orientation preserving.The prescription of [ BM04 ] to construct such systems is now as follows.(1) For any trivial bundle E over ˙Σ = C with ∞ as a negative puncture and anyasymptotic operator A ∞ , choose an arbitrary continuous family of orien-tations for the operators in C R ( E, { A ∞ } ), subject only to the requirementthat these should match the complex orientation whenever A ∞ is complexlinear.(2) For any trivial bundle E − over ˙Σ = C with ∞ as a positive puncture, anyasymptotic operator A ∞ and any D − ∈ C R ( E − , { A ∞ } ), let E + denote thetrivial bundle over C with a negative puncture as in step (1), choose any D + ∈ C R ( E + , { A ∞ } ) and construct the resulting family of glued operators D R ∈ C R ( E R ) , where the E R are trivial bundles over S . Since S has no punctures, D R has a natural complex orientation, so define the orientation of D − to bethe one that is compatible via Lemma 11.13 with this and the orientationchosen for D + in step (1).(3) For an arbitrary ( E, J ) → ( ˙Σ , j ), glue positive and negative planes to ˙Σ toproduce a bundle over a closed surface b Σ, and define the orientation of any D ∈ C R ( E, { A z } z ∈ Γ ) to be compatible via Lemma 11.13 with the choicesin steps (1) and (2) and the complex orientation for operators over b Σ.It should be easy to convince yourself that if we now vary the bundle (
E, J ) → ( ˙Σ , j ) or the operators on this bundle (but not the asymptotic operators!) contin-uously, the capping procedure described in step (3) above produces a continuousfamily of Cauchy-Riemann type operators on bundles over closed Riemann surfaces.Since these all carry the complex orientation, the resulting orientations of the orig-inal operators vary continuously. It is similarly clear from the construction thatany Cauchy-Riemann operator whose asymptotic operators are all complex linearwill end up with the complex orientation. Bourgeois and Mohnke use this fact toprove that any system of orientations constructed in this way is compatible with all possible linear gluing maps arising from Lemma 11.13. The idea is to reduceit to the complex-linear case by gluing cylinders to the ends of any asymptoticallyHermitian bundle so that the asymptotic operators can be changed at will; see[ BM04 , Proposition 8].
The heuristic proofs in § D ∈ C R ( E, { A z } z ∈ Γ ), and D ′ is the same operator after interchangingtwo of the punctures in Γ. Imagine gluing ( E, J ) → ( ˙Σ , j ) to trivial bundles E and E over planes in order to cap off the two punctures that are being interchanged,and choose Cauchy-Riemann operators D and D on these planes to form a gluedoperator on the capped surface. This capping procedure is done one plane at a time, ectures on Symplectic Field Theory and the order of the two punctures determines which plane is glued first. Compati-bility with the isomorphisms of Lemma 11.13 then dictates that the orientations ofdet( D ) and det( D ′ ) match if and only if the orientations of det( D ) ⊗ det( D ) anddet( D ) ⊗ det( D ) match. Since orientations of det( D i ) for i = 1 , D i ⊕ coker D i , reversing the order of the tensor product changesorientations if and only if both of these direct sums are odd dimensional, whichmeans ind( D ) and ind( D ) are both odd. If the bundles have complex rank n andthe asymptotic operators are A i for k = 1 ,
2, we haveind( D i ) = nχ ( C ) ± µ CZ (( − i∂ t ⊕ A i ) ± δ ) = n − ± µ CZ ( A i ) , which matches n − µ CZ ( A i ) modulo 2. This proves Proposition 11.4.Similarly for Proposition 11.5, we consider the action of the generator ψ ∈ Z m on det( D ) where ψ rotates the cylindrical end by 1 /m at some puncture where thetrivialized asymptotic operator A is of the form − i∂ t − S ( mt ) for a loop of symmetricmatrices S ( t ). Capping off this puncture with a plane carrying a Cauchy-Riemannoperator D ∞ , coherence dictates that the same transformation must act the sameway on the orientation of det( D ∞ ). Since ψ m = 1, ψ cannot reverse this orientationif m is odd. To understand the case of m even, note first that we are free tochoose D ∞ so that it is an m -fold cover, meaning it is related to the branched cover ϕ : C → C : z z m by D ∞ ( η ◦ ϕ ) = ϕ ∗ b D ∞ η for some other Cauchy-Riemann operator b D ∞ , which is asymptotic to ˆ A := − i∂ t − S ( t ). Now the group Z m generated by ψ acts on ker D ∞ and coker D ∞ , so represen-tation theory tells us ker D ∞ = V ⊕ V − ⊕ V rot coker D ∞ = W ⊕ W − ⊕ W rot , where ψ acts on V ± and W ± as ± and acts as orientation-preserving rotations on V rot and W rot . It follows that ψ reverses the orientation of ker D ∞ ⊕ coker D ∞ if andonly if dim V − − dim W − is odd. Now observe that there are natural isomorphisms V = ker b D ∞ , W = coker b D ∞ , hence dim V − − dim W − = ind( D ∞ ) − ind( b D ∞ ) (mod 2) . This difference in Fredholm indices is precisely µ CZ ( A ) − µ CZ ( ˆ A ) up to a sign, andthis completes the proof of Proposition 11.5.ECTURE 12 The generating function of SFTContents H ( M ) has torsion? 26312.7.2. Combinatorial conventions 26312.7.3. Coefficients: Q , Z or Z ? 264 It is time to begin deriving algebraic consequences from the analytical results ofthe previous lectures. We saw the simplest possible example of this in Lecture 10,where the behavior of holomorphic cylinders in symplectizations of contact mani-folds without contractible Reeb orbits led to a rudimentary version of cylindricalcontact homology HC ∗ ( M, ξ ) with Z coefficients. Unfortunately, the condition oncontractible orbits means that this version of HC ∗ ( M, ξ ) cannot always be defined,and even when it can, it only counts cylinders—we would only expect it to capturea small fragment of the information contained in more general moduli spaces ofholomorphic curves. Extracting information from these general moduli spaces willrequire enlarging our algebraic notion of what a Floer-type theory can look like.
For this and the next lecture, we fix the following fantastically optimistic as-sumption:
Assumption . One can choose suitably compatible almostcomplex structures so that all pseudoholomorphic curves are Fredholm regular.
This assumption held in Lecture 10 for the curves we were interested in, becausethey were all guaranteed for topological reasons to be somewhere injective. It canalso be shown to hold under some very restrictive conditions on Conley-Zehnderindices in dimension three, see [
Nel15 , Nel13 ]. Both of those are very lucky situ-ations, and as we’ve discussed before, the assumption cannot generally be achievedmerely by perturbing J generically—it must sometimes fail for curves that are mul-tiply covered, and such curves always exist (see § Chris Wendl way in reality to ensure something like Assumption 12.1 is to perturb the nonlin-ear Cauchy-Riemann equation more abstractly, e.g. by replacing ¯ ∂ J u = 0 with aninhomogeneous equation of the form ¯ ∂ J u = ν for a generic perturbation ν . This is the standard technique in certain versions ofGromov-Witten theory, see e.g. [ RT95 , RT97 ]. Alternatively, one can allow J todepend generically on points in the domain rather than just points in the target,as in [ MS04 , § M ( J ) and the strata of its compactification M ( J ). As observedin [ Sal99 , § M ( J ) makes it necessaryfor any sufficiently general abstract perturbation scheme to involve multivalued per-turbations, and it is important for these perturbations to be “coherent” in a senseanalogous to our discussion of orientations in the previous lecture. These notionshave not yet all been developed in a sufficiently consistent and general way to givea rigorous definition of SFT, though there has been much progress: this is the mainobjective of the long-running polyfold project by Hofer-Wysocki-Zehnder [ Hof06 ].Recently, a quite different and much more topological approach has been proposedby John Pardon [
Par ].For most of this lecture we will ignore these subtleties and simply adopt As-sumption 12.1 as a convenient fiction, thus pretending that all components of M ( J )are smooth orbifolds of the correct dimension and all gluing maps are smooth. All“theorems” stated under this assumption should be read with the caveat that theyare only true in a fictional world in which the assumption holds. Even if it is afiction, one can get quite far with this point of view: it is still possible not onlyto deduce the essential structure of what we assume will someday be a rigorouslydefined polyfold-based SFT, but also to infer the existence of certain contact in-variants that have interesting rigorous applications requiring only well-establishedtechniques, e.g. the cobordism obstructions discovered in [ LW11 ]. The goal is to define an invariant of closed (2 n − M, ξ ) with closed nondegenerate Reeb orbits as generators and a Floer-typedifferential counting J -holomorphic curves in the symplectization ( R × M, d ( e r α )).The auxiliary data we choose must obviously therefore include a nondegenerate con-tact form α and a generic J ∈ J ( α ), for which we shall assume Assumption 12.1holds. For convenience, we will also assume throughout most of this lecture: Assumption . H ( M ) is torsion free. This is needed mainly in order to be able to define an integer grading, thoughwithout this assumption, it is still always possible to define a Z -grading—see § α, J ) with the following additional choices: ectures on Symplectic Field Theory (1) Coherent orientations as in Lecture 11 for the moduli spaces M $ ( J ) withasymptotic markers.(2) A collection of reference curves S ∼ = C , . . . , C r ⊂ M whose homology classes form a basis of H ( M ).(3) A unitary trivialization of ξ along each of the reference curves C , . . . , C r ,denoted collectively by τ .(4) A spanning surface C γ for each periodic Reeb orbit γ : this is a smoothmap of a compact and oriented surface with boundary into M such that ∂C γ = X i m i [ C i ] − [ γ ]in the sense of singular 2-chains, where m i ∈ Z are the unique coefficientswith [ γ ] = P i m i [ C i ] ∈ H ( M ).These choices determine the following. To any collections of Reeb orbits γ ± =( γ ± , . . . , γ ± k ± ) and any relative homology class A ∈ H ( M, ¯ γ + ∪ ¯ γ − ) with ∂A = P i [ γ + i ] − P j [ γ − j ], we can now associate a cycle in absolute homology, A + X i C γ + i − X j C γ − j ∈ H ( M ) . Indeed, the boundary of this real 2-chain is a sum of linear combinations of thereference curves C i , which add up to zero because P i [ γ + i ] and P j [ γ − j ] are homolo-gous. We shall abuse notation and use this correspondence to associate the absolutehomology class [ u ] ∈ H ( M )to any asymptotically cylindrical holomorphic curve u in R × M . Adapting theprevious notation, M g,m ( J, A, γ + , γ − )for A ∈ H ( M ) will now denote a moduli space of curves whose relative homologyclasses glue to the chosen capping surfaces to form A .Secondly, the chosen trivializations τ along the reference curves can be pulledback and extended over every capping surface C γ , giving trivializations of ξ alongevery orbit γ uniquely up to homotopy. We shall define µ CZ ( γ ) ∈ Z from now on to mean the Conley-Zehnder index of γ relative to this trivialization. Exercise . Show that if H ( M ) has no torsion and u : ˙Σ → R × M isasymptotically cylindrical, then its relative first Chern number with respect to thetrivializations τ described above satisfies c τ ( u ∗ T ( R × M )) = c ([ u ]) , where c ([ u ]) denotes the evaluation of c ( ξ ) ∈ H ( M ) on [ u ] ∈ H ( M ). Chris Wendl
By Exercise 12.3, the index of a curve u : ( ˙Σ = Σ \ Γ , j ) → ( R × M, J ) with[ u ] = A ∈ H ( M ) and asymptotic orbits { γ z } z ∈ Γ ± can now be written as(12.1) ind( u ) = − χ ( ˙Σ) + 2 c ( A ) + X z ∈ Γ + µ CZ ( γ z ) − X z ∈ Γ − µ CZ ( γ z ) . In order to keep track of homology classes of holomorphic curves algebraically,we can define our theory to have coefficients in the group ring Q [ H ( M )], or moregenerally, R := Q [ H ( M ) /G ]for a given subgroup G ⊂ H ( M ). Elements of R will be written as finite sums X i c i e A i ∈ R, c i ∈ Q , A i ∈ H ( M ) /G, where the multiplicative structure of the group ring is derived from the additivestructure of H ( M ) /G by e A e B := e A + B . The most common examples of G are H ( M ) and the trivial subgroup, giving R = Q or R = Q [ H ( M )] respectively. Wewill see a geometrically meaningful example in between these two extremes in thenext lecture.Finally, we define certain formal variables which have degrees in Z or Z N forsome N ∈ N , and will serve as generators in our graded algebra. To each closedReeb orbit γ we associate two variables, q γ , p γ , whose integer-valued degrees are | q γ | = n − µ CZ ( γ ) , | p γ | = n − − µ CZ ( γ ) . To remember these numbers, think of the index of a J -holomorphic plane u positivelyor negatively asymptotic to γ , with [ u ] = 0.We also assign an integer grading to the group ring Q [ H ( M )] such that rationalnumbers have degree 0 and | e A | = − c ( A ) , for A ∈ H ( M ) . If c ( A ) = 0 for every A ∈ G , in particular if c ( ξ ) = 0, then this descends to aninteger grading on the ring R = Q [ H ( M ) /G ]. Otherwise, R inherits a Z N -grading,where N := min (cid:8) c ( A ) > (cid:12)(cid:12) A ∈ G (cid:9) . A Z -grading is well defined in every case.The algebra will include one additional formal variable ~ , which is defined tohave degree | ~ | = 2( n − . The degrees of ~ and the p γ and q γ variables should all be interpreted modulo 2 N if c ( ξ ) | G = 0.The algebra of SFT uses monomials in the variables p γ and q γ respectively toencode sets of positive and negative asymptotic orbits of holomorphic curves, whilethe group ring R = Q [ H ( M ) /G ] is used to keep track of the homology classes ofsuch curves, and powers of ~ are used to keep track of their genus. More precisely, ectures on Symplectic Field Theory given g ≥ A ∈ H ( M ) and ordered lists of Reeb orbits γ ± = ( γ ± , . . . , γ ± k ± ), weencode the moduli space M g, ( J, A, γ + , γ − ) formally via the product(12.2) e A ~ g − q γ − p γ + := e A ~ g − q γ − . . . q γ − k − p γ +1 . . . p γ + k + , where we are abusing notation by identifying A with its equivalence class in H ( M ) /G if G is nontrivial. Notice that according to the above definitions, this expression hasdegree | e A ~ g − q γ − p γ + | = | e A | + ( g − | ~ | + k − X i =1 (cid:2) ( n −
3) + µ CZ ( γ − i ) (cid:3) + k + X i =1 (cid:2) ( n − − µ CZ ( γ + i ) (cid:3) = − c ( A ) + (2 g − k + + k − )( n − − k + X i =1 µ CZ ( γ + i ) + k − X i =1 µ CZ ( γ − i )= − vir-dim M g, ( J, A, γ + , γ − ) , (12.3)interpreted modulo 2 N if c ( ξ ) | G = 0. The orientation results in Lecture 11 suggestintroducing a supercommutativity relation for the variables q γ and p γ : defining thegraded commutator bracket by(12.4) [ F, G ] :=
F G − ( − | F || G | GF, we define a relation on the set of all monomials of the form q γ − p γ + by setting(12.5) [ q γ , q γ ] = [ p γ , p γ ] = 0for all pairs of orbits γ and γ . As a consequence, permuting the orbits in the lists γ ± changes the sign of the monomial (12.2) if and only if it changes the orientationof the corresponding moduli space. In particular, any product that includes multiplecopies of an odd generator q γ or p γ is identified with 0. This accounts for the fact thatany rigid moduli space M g, ( J, A, γ + , γ − ) with two copies of γ among its positive ornegative asymptotic orbits contains zero curves when counted with the correct signs:every curve is cancelled by a curve that looks identical except for a permutation oftwo of its punctures. To write down the SFT generating function, let M σ ( J ) := M ( J ) (cid:14) ∼ denote the space of equivalence classes where two curves are considered equivalentif they have parametrizations that differ only in the ordering of the punctures. Thisspace is in some sense more geometrically natural than M ( J ) or M $ ( J ), but due tothe orientation results in the previous lecture, less convenient for technical reasons. Chris Wendl
Given u : ( ˙Σ , j ) → ( R × M, J ) representing a nonconstant element of M σ ( J ) withno marked points, it is natural to defineAut σ ( u ) ⊂ Aut(Σ , j )as the (necessarily finite) group of biholomorphic transformations ϕ : (Σ , j ) → (Σ , j )satisfying u = u ◦ ϕ ; in particular, elements of Aut σ ( u ) are allowed to permute thepunctures, so Aut σ ( u ) is generally a larger group than the usual Aut( u ). For k ∈ Z ,let M σk ( J ) ⊂ M σ ( J )denote the subset consisting of index k curves that have no marked points and whoseasymptotic orbits are all good (see Definition 11.6 in Lecture 11).We now define the SFT generating function as a formal power series(12.6) H = X u ∈M σ ( J ) / R ǫ ( u ) | Aut σ ( u ) | ~ g − e A q γ − p γ + , where the terms of each monomial are determined by u ∈ M σ ( J ) as follows: • g is the genus of u ; • A is the equivalence class of [ u ] ∈ H ( M ) in H ( M ) /G ; • γ ± = ( γ ± , . . . , γ ± k ± ) are the asymptotic orbits of u after arbitrarily fixingorderings of its positive and negative punctures; • ǫ ( u ) ∈ { , − } is determined by the chosen coherent orientations on M $ ( J ).Specifically, given the chosen ordering of the punctures and an arbitrarychoice of asymptotic markers at each puncture, u determines a 1-dimensionalconnected component of M $ ( J ), and we define ǫ ( u ) = +1 if and only if thecoherent orientation of M $ ( J ) matches its tautological orientation deter-mined by the R -action.Note that while both ǫ ( u ) and the corresponding monomial q γ − p γ + depend on achoice of orderings of the punctures, their product does not depend on this choice.Moreover, ǫ ( u ) does not depend on the choice of asymptotic markers since curveswith bad asymptotic orbits are excluded from M σ ( J ). Since every monomial in H corresponds to a holomorphic curve of index 1, (12.3) implies | H | = − . There are various combinatorially more elaborate ways to rewrite H . For anyReeb orbit γ , let κ γ := cov( γ ) ∈ N denote its covering multiplicity, and for a finite list of orbits γ = ( γ , . . . , γ k ), let κ γ := k Y i =1 κ γ i . Given u ∈ M σ ( J ) with k ± ≥ γ ± = ( γ ± , . . . , γ k ± ± ), there are k + ! k − ! κ γ + κ γ − ways to order the puncturesand choose asymptotic markers, but some of them are equivalent since (by an easy ectures on Symplectic Field Theory variation on Proposition 11.1) the finite group Aut σ ( u ) acts freely on this set ofchoices. As a result, (12.6) is the same as(12.7) H = X u ∈M $1 ( J ) / R ǫ ( u ) k + ! k − ! κ γ + κ γ − ~ g − e A q γ − p γ + , where M $1 ( J ) denotes the space of all index 1 curves without marked points in M $ ( J ), and the rest of the mononomial is determined by the condition that u belongs to M $ g, ( J, A, γ + , γ − ), with no need for any arbitrary choices. Another wayof writing this is(12.8) H = X g,A, γ + , γ − (cid:0) M $ g, ( J, A, γ + , γ − ) (cid:14) R (cid:1) k + ! k − ! κ γ + κ γ − ~ g − e A q γ − p γ + , where the sum ranges over all integers g ≥
0, homology classes A ∈ H ( M ) andordered tuples of Reeb orbits γ ± = ( γ ± , . . . , γ ± k ± ), and (cid:0) M $ g, ( J, A, γ + , γ − ) (cid:14) R (cid:1) ∈ Z is the signed count of index 1 connected components in M $ g, ( J, A, γ + , γ − ). Forfixed g and γ ± , the union of these spaces for all A ∈ H ( M ) is finite due to SFTcompactness, as the energy of curves in ( R × M, d ( e t α )) is computed by integratingexact symplectic forms and thus (by Stokes) admits a uniform upper bound in termsof γ + . For this reason, (12.8) defines a formal power series in the p variables andin ~ , with coefficients that are polynomials in the q variables and the group ring R .We played a slightly sneaky trick in writing down (12.7) and (12.8): these sum-mations to not exclude bad orbits, whereas (12.6) was a sum over curves u that arenot asymptotic to any bad orbits—a necessary exclusion in that case because ǫ ( u )would otherwise depend on choices of asymptotic markers. The reason bad orbitsare allowed in (12.8) is that their total contribution adds up to zero: indeed, badorbits are always multiple covers with even multiplicity, so whenever u ∈ M $ ( J )has a puncture approaching a bad orbit with multiplicity 2 m , there are exactly2 m − M $ ( J ) that differ only by adjustment of the marker atthat one puncture, and by Proposition 11.5, half of these cancel out the other halfin the signed count. We’ve already seen that a similar remark explains the harmlessabsence from (12.8) of terms with multiple factors of any odd generator q γ or p γ . Remark . Readers famliar with Floer homology may see a resemblancebetween the group ring R = Q [ H ( M ) /G ] and the Novikov rings that often appearin Floer homology, though R is not a Novikov ring since it only allows finite sums.In Floer homology, the Novikov ring sometimes must be included because counts ofcurves may fail to be finite, though they only do so if the energies of those curvesblow up. The situation above is somewhat different: since the symplectization is anexact symplectic manifold, Stokes’ theorem implies that energy cannot blow up ifthe positive asymptotic orbits are fixed, and one therefore obtains well-defined curvecounts no matter the choice of the coefficient ring R . The use of the group ring isconvenient however for two reasons: first, without it one cannot always define aninteger grading, and second, different choices of coefficients can sometimes be used Chris Wendl to detect different geometric phenomena via SFT. We will see an example of thelatter in Lecture 13.The compactness and gluing theory of SFT is encoded algebraically by viewing H as an element on a noncommutative operator algebra determined by the commutatorrelations [ p γ , q γ ] = κ γ ~ [ p γ , q γ ′ ] = 0 if γ = γ ′ . (12.9)Here [ , ] again denotes the graded commutator (12.4), so “commuting” generatorsactually anticommute whenever they are both odd. The rest of the multiplicativestructure of this algebra is determined by requiring all elements of R and powersof ~ (all of which are even generators) to commute with everything, meaning alloperators are R [[ ~ ]]-linear.One concrete representation of this operator algebra is as follows: let A denotethe graded supercommutative unital algebra over R generated by the set (cid:8) q γ (cid:12)(cid:12) γ a good Reeb orbit (cid:9) . The ring of formal power series A [[ ~ ]] is then an R [[ ~ ]]-module. Define each of thegenerators q γ to be R [[ ~ ]]-linear operators on A [[ ~ ]] via multiplication from the left,and define p γ : A [[ ~ ]] → A [[ ~ ]] by(12.10) p γ = κ γ ~ ∂∂q γ . Here the R [[ ~ ]]-linear partial derivative operator is defined via ∂∂q γ q γ = 1 , ∂∂q γ q γ ′ = 0 for γ = γ ′ and the graded Leibniz rule ∂∂q γ ( F G ) = ∂F∂q γ G + ( − | q γ || F | F ∂G∂q γ for all homogeneous elements F, G ∈ A [[ ~ ]]. Exercise . Check that the operator p γ : A [[ ~ ]] → A [[ ~ ]] defined above hasthe correct degree and satisfies the commutation relations (12.5) and (12.9).Notice that while H contains terms of order − ~ , every term also containsat least one p γ variable since all index 1 holomorphic curves in ( R × M, d ( e t α )) haveat least one positive puncture. The substitution (12.10) thus produces a differentialoperator in which every term contains a nonnegative power of ~ , giving a well-defined R [[ ~ ]]-linear operator D SFT : A [[ ~ ]] H −→ A [[ ~ ]] . The following may be regarded as the fundamental theorem of SFT.
Theorem . H = 0 . ectures on Symplectic Field Theory We will discuss in § H SFT ∗ ( M, ξ ) := H ∗ ( A [[ ~ ]] , D SFT ) , which will turn out to be an invariant of ( M, ξ ) in the sense that any two choicesof α , J and the other auxiliary data described in § A [[ ~ ]] isan algebra, its product structure does not descend to H SFT ∗ ( M, ξ ) since D SFT is not aderivation—indeed, it is a formal sum of differential operators of all orders, not justorder one. In the next lecture we will discuss various ways to produce homologicalinvariants out of H with nicer algebraic structures.On the other hand, it is fairly easy to understand the geometric meaning of thecomplex ( A [[ ~ ]] , D SFT ) in Floer-theoretic terms. Each individual curve u ∈ M σ ( J )with genus g , homology class A ∈ H ( M ) and asymptotic orbits γ ± = ( γ ± , . . . , γ ± k ± )contributes to D SFT the differential operator ǫ ( u ) | Aut σ ( u ) | κ γ + ~ g + k + − e A q γ − . . . q γ − k − ∂∂q γ +1 . . . ∂∂q γ + k + . Applying this operator to a monomial q γ . . . q γ m ∈ A [[ ~ ]] that does not contain all ofthe generators q γ +1 , . . . , q γ + k + will produce zero, and its effect on a product that doescontain all of these generators will be to eliminate them and multiply q γ − . . . q γ − k − bywhatever remains, plus some combinatorial factors and signs that may arise fromdifferentiating by the same q γ more than once. Ignoring the combinatorics andsigns for the moment, this operation on q γ . . . q γ m has a geometric interpretation:it counts all potentially disconnected J -holomorphic curves of index 1 (i.e. disjointunions of u with trivial cylinders) that have γ , . . . , γ m as their positive asymptoticorbits; see Figure 12.1. In other words, the action of D SFT on each monomial q γ for γ = ( γ , . . . , γ m ) is determined by a formula of the form(12.11) D SFT q γ = ∞ X g =0 X A ∈ H ( M ) X γ ′ m X k =1 ~ g + k − e A n g ( γ , γ ′ , k ) q γ ′ , where n g ( γ , γ ′ , k ) is a product of some combinatorial factors with a signed count ofgenerally disconnected index 1 holomorphic curves of genus g and homology class A with positive ends at γ and negative ends at γ ′ , such that the nontrivial connectedcomponent has exactly k positive ends. The presence of the combinatorial factorshidden in n g ( γ , γ ′ , k ) is a slightly subtle point which we will try to clarify in thefollowing sections. As in all versions of Floer theory, the proof that H = 0 is based on the fact thatcertain moduli spaces are compact oriented 1-dimensional manifolds with bound-ary, and the signed count of their boundary points is therefore zero. We must be Chris Wendl γ γ γ γ γ γ γ γ u Figure 12.1.
Counting disjoint unions of index 1 curves u ∈M , ( J, A, ( γ , γ , γ ) , ( γ , γ )) with some trivial cylinders contributesa multiple of ~ e A q γ q γ q γ q γ q γ to D SFT ( q γ q γ q γ q γ q γ q γ ).careful of course because, strictly speaking, M ( J ) is not a manifold even whenAssumption 12.1 holds—it is an orbifold, with the possibility of singularities at mul-tiply covered curves with nontrivial automorphism groups. On the other hand, onecan show that (after excluding curves with bad asymptotic orbits) it is an oriented orbifold, and oriented 1-dimensional orbifolds happen to be very simple objects:since smooth finite group actions on R cannot be nontrivial without reversing ori-entation, all oriented 1-dimensional orbifolds are actually manifolds, suggesting thesimple formula “ ∂ M ( J ) = 0.”I have placed this formula in quotation marks for a reason. The reality of thesituation is somewhat more complicated.This is in fact where it becomes important to remember that Assumption 12.1, inthe way that we stated it, really is not just science fiction but fantasy : transversalityis sometimes impossible to achieve for multiple covers, and we must therefore at leasthave a sensible back-up plan for such cases. To see the problem, remember that ourlocal structure theorem for M ( J ) was proved by identifying it in a neighborhood ofany curve u : ( ˙Σ , j ) → ( R × M, J ) with a set of the form¯ ∂ − J (0) (cid:14) G, where ¯ ∂ J : T ×B k,p,δ → E k − ,p,δ is a smooth section of a Banach space bundle E k − ,p,δ over the product of a Teichm¨uller slice T through j with a Banach manifold B k,p,δ of maps ˙Σ → R × M , and G is the group of automorphisms of j , whose action onthe base G × ( T × B k,p,δ ) → T × B k,p,δ : ( ψ, ( j, u )) ( ψ ∗ j, u ◦ ψ )preserves ¯ ∂ − J (0). In fact, the action of G on T × B k,p,δ is covered by a natural actionon the bundle E k − ,p,δ , and the reason for it preserving the zero-set is that ¯ ∂ J is anequivariant section, ¯ ∂ J ( ψ ∗ j, u ◦ ψ ) = ψ ∗ ¯ ∂ J ( j, u ) . As you may know if you’ve ever heard a talk about polyfolds, there are some analyticalproblems with this discussion if G is a Lie group of positive dimension: its action on the infinite-dimensional manifold B k,p,δ of non-smooth maps cannot then be considered smooth in any con-ventional sense. This problem leads to the introduction of sc-smooth structures, cf. [ HWZ07 ].There is no problem however if G is finite, e.g. if the underlying Riemann surface is stable, whichwe may as well assume for this discussion. ectures on Symplectic Field Theory If G is finite, then another way to say this is that ¯ ∂ J is a smooth Fredholm sectionof the infinite-dimensional orbibundle E k − ,p,δ /G over the orbifold ( T × B k,p,δ ) /G ,whose isotropy group at ( j , u ) is Aut( u ). This section is transverse to the zero-section if and only if the usual regularity condition holds, making ¯ ∂ − J (0) /G a suborb-ifold of ( T × B k,p,δ ) /G whose isotropy group at ( j , u ) is some quotient of Aut( u ). Remark . Most sensible definitions of the term orbifold (cf. [
ALR07 , Dav , FO99 ]) require local models of the form U /G , where U is a G -invariant open subsetof a vector space on which the finite group G acts smoothly and effectively —thelatter condition is necessary in order to have isotropy groups that are well-definedup to isomorphism at every point. In the above example, G acts effectively on T × B k,p,δ but might have a nontrivial subgroup H ⊂ G of transformations thatfix every element of ¯ ∂ − J (0), in which case the G -action on ¯ ∂ − J (0) can be replacedby an effective action of G/H . The isotropy group of ( j , u ) ∈ ¯ ∂ − J (0) /G is thenAut( u ) / (Aut( u ) ∩ H ).Now to see just how unreasonably optimistic Assumption 12.1 is, notice that it’seasy to think up examples of smooth orbibundles in which zeroes of sections can never be regular if they have nontrivial isotropy. Example . Let M = C / Z with Z acting as the antipodal map, andconsider the trivial complex line bundle E = M × C = ( C × C ) / Z , where the Z action on C × C identifies ( z, v ) with ( − z, v ). A smooth function f : C → C thenrepresents a section of the orbibundle E → M if and only if f ( z ) = f ( − z ) for all z .This implies that if f (0) = 0, then df (0) = 0. It is possible to perturb f genericallyto a section that is transverse to the zero-section, but such a perturbation can neverhave zeroes at 0.Of course, we do know how to assign Z -valued orders to degenerate zeroes ofsections, e.g. f ( z ) = z defines a section of E → M with a zero of order 2 at 0.Notice however that if we perturb this to f ǫ ( z ) = z + ǫ for ǫ > f ǫ hastwo simple zeroes at points near the origin, but they are actually the same point in C / Z , giving a count of only 1 zero. This means that if we give the zero of f at theorigin its full weight, then we are counting wrongly—the resulting count will not behomotopy invariant. The correct algebraic count of zeroes is evidently(12.12) f − (0) := X z ∈ f − (0) ⊂ M ord( f ; z ) κ z ∈ Q , where ord( f ; z ) ∈ Z is the order of the zero (computed in the usual way as a windingnumber, or in higher dimensions as the degree of a map of spheres, cf. [ Mil97 ]), and κ z ∈ N denotes the order of the isotropy group at z . Exercise . Convince yourself that for any smooth oriented orbibundle E → M of real rank m over a compact, smooth and oriented m -dimensional orbifold M without boundary, the count (12.12) gives the same result for any section withisolated zeroes. Hint: The space of sections of an orbibundle is still a vector space, so any two If you’re still not sure what an orbibundle is, a definition can be found in [
FO99 , Chapter 1]. Chris Wendl are homotopic. Since M and [0 , are both compact, it suffices to focus on smallperturbations of a single section on a single orbifold chart. For a slightly different perspective on (12.12), consider the special case of a closedorbifold that is the quotient of a closed manifold f M by an effective orientation-preserving finite group action, M = f M /G.
Suppose e E → f M is an oriented vector bundle with rank equal to dim M , and G alsoacts on e E by orientation-preserving linear bundle maps that cover its action on f M ,so the quotient E = e E/G → M is an orbibundle. A section f : M → E is then equivalent to a G -equivariant section˜ f : f M → e E , and the signed count of zeroes f − (0) = X z ∈ ˜ f − (0) ⊂ f M ord( ˜ f ; z ) ∈ Z is of course the same for any section that has only isolated zeroes. It can also beexpressed in terms of f since any z ∈ f − (0) ⊂ M has exactly | G | /κ z lifts to pointsin ˜ f − (0) ⊂ f M , implying f − (0) = X z ∈ f − (0) ⊂ M | G | κ z ord( f ; z )and thus f − (0) = | G | f − (0). The invariance of (12.12) is now an immediateconsequence of the invariance of f − (0), which follows from the standard argumentas in [ Mil97 ].Now, if you enjoyed reading [
Mil97 ] as much as I did, then it may seem temptingto try proving invariance of (12.12) in general by choosing a generic homotopy H : [0 , × M → E between two generic sections f and f and showing that H − (0) ⊂ [0 , × M is a compact oriented 1-dimensional orbifold with boundary.As we observed at the beginning of this section, H − (0) is then actually a manifold,so the signed count of its boundary points should be zero. But this would givethe wrong result: it would suggest that P z ∈ f − (0) ⊂ M ord( f ; z ) should be homotopyinvariant, without the rational weights, and we’ve already seen that this is not true.What is going on here? The answer is that the homogopy H cannot in general bemade transverse to the zero-section, now matter how generically we perturb it! Itis an illustration of the fundamental conflict between the notions of genericity and equivariance . Example . Let M = C / Z as in Example 12.8, but define the complexorbibundle E → M by E = ( C × C ) (cid:14) ( z, v ) ∼ ( − z, − v ) , i.e. the Z -action also acts antipodally on fibers. Now a smooth function f : C → C defines a section of E if and only if f ( − z ) = − f ( z ), hence all such sections have a ectures on Symplectic Field Theory zero at the origin. Compare the two sections f ( x + iy ) = x + iy, f ( x + iy ) = ( x − x ) + iy. They have qualitatively the same behavior near infinity, meaning in particular thatthey are homotopic through a family of sections whose zeroes are confined to somecompact subset, thus we expect the algebraic count of zeroes to be the same for both.This is true if the count is defined by (12.12): we have f − (0) = f − (0) = , inparticular the negative zero of f at the origin counts for − / , ∼ ( − ,
0) counts for 1. We see that the inclusion of the rational weights κ x is crucial for this result. Notice that if H : [0 , × M → E is a homotopy ofsections from f to f , then H ( τ,
0) = 0 for all τ , thus ∂ τ H ( τ,
0) vanishes and dH ( τ,
0) = df τ (0)where f τ = H ( τ, · ). But df τ (0) cannot be an isomorphism for all τ ∈ (0 ,
1) since df (0) preserves orientation while df (0) reverses it. This is not a problem that canbe fixed by making H more generic—the homotopy will never be transverse to thezero-section, no matter what we do.The need to address issues of the type raised by the above examples leads natu-rally to the notion of multisections as outlined in [ Sal99 , §
5] and [
FO99 ], and thisis a major feature of the analysis under development by Hofer-Wysocki-Zehnder, seefor example [
HWZ10 ]. In Example 12.10 for instance, one can consider functions f : C → Sym ( C ) := ( C × C ) (cid:14) ( z , z ) ∼ ( z , z ) , which can be regarded as doubly-valued sections of E → M if f is Z -equivariantfor the antipodal action of Z on the symmetric product Sym ( C ). Such a sectionis considered single-valued at any point z where f ( z ) is of the form [( v, v )], so onecan now imagine homotopies from f to f through doubly-valued sections. Oneadvantage of this generalization is that f can now take nonzero values of the form[( v, − v )] at the origin, e.g. if g : C → C is any odd function, then f ( z ) := [( g ( z ) + c, g ( z ) − c )]is a well-defined multisection for every c ∈ C . Exercise . Find a homotopy between the sections f and f of Exam-ple 12.10 through doubly-valued sections, such that the homotopy is transverse tothe zero-section.You may notice if you work out Exercise 12.11 that the zero set of the homotopyin [0 , × M is still not submanifold or suborbifold. Instead, it naturally carriesthe structure of a weighted branched manifold with boundary . The rational weightsattached to every point in this object can be used to explain the weights appearingin (12.12) and thus give a Milnor-style proof that f − (0) ∈ Q is invariant.We will not discuss multisections or weighted branched manifolds any further,but the main takeaway from this discussion should be that the “right” way to count0-dimensional orbifolds algebraically is always some version of (12.12), and the countin general is a rational number, not an integer. We’ve discussed this above fromthe perspective of obtaining a homotopy-invariant count, but the same logic applies Chris Wendl to any Floer-type theory since the relation ∂ = 0 is typically based on similararguments via 1-dimensional moduli spaces with boundary. While a more simplisticnotion of counting may produce well-defined homology theories in isolated caseswhere Assumption 12.1 holds (e.g. in [ Nel15 ]), we cannot expect it to generalizebeyond these cases, due to the fundamental conflict between transversality andequivariance. On the other hand, it will be possible in our situation to removeisotropy from the picture by lifting to moduli spaces with asymptotic markers; themoduli space we’re interested in is always the quotient of this larger space by afinite group action, so the situation is analogous to replacing an orbibundle E = e E/G → f M /G by an ordinary vector bundle e E over a manifold f M . In the infinite-dimensional setting, transversality is still a hard problem, but having lifted to amanifold and thus removed the need for equivariance, there is no longer any a priori reason why it cannot be solved by choosing sufficiently generic perturbations. Thismakes counting curves with rational weights seem a much more promising methodfor defining invariants, and we will adopt this perspective in the discussion to follow. Under an extra assumption on the complex ( A [[ ~ ]] , D SFT ), we can recover fromit a more general version of the cylindrical contact homology we saw in Lecture 10.Suppose in particular that there are no index 1 holomorphic planes in R × M , soevery term in ~ H has at least one factor of either ~ or one of the q γ variables. Then D SFT = X γ,γ ′ ,A κ γ X u ∈M , ( J,A,γ,γ ′ ) / R ǫ ( u ) | Aut( u ) | e A q γ ′ ∂∂q γ + . . . , where the first sum is over all pairs of good Reeb orbits γ and γ ′ , and the ellipsisis a sum of terms that all include at least a positive power of ~ or two q γ variablesor two partial derivatives. Let us abbreviate the spaces M , ( J, A, γ, γ ′ ) / R of R -equivalence classes of J -holomorphic cylinders by M A ( γ, γ ′ ), and notice that forany u ∈ M A ( γ, γ ′ ), the automorphism group is a cyclic group of order equal to thecovering multiplicity | Aut( u ) | = κ u := cov( u ) ∈ N . Thus for any single generator q γ , we have D SFT q γ = ∂ CCH q γ + O ( | q | , ~ ) , where(12.13) ∂ CCH q γ := κ γ X γ ′ ,A X u ∈M A ( γ,γ ′ ) ǫ ( u ) κ u e A q γ ′ . The fact that D = 0 thus implies ∂ = 0 , and the homology of the graded R -module generated by { q γ | γ good } with differ-ential ∂ CCH is an obvious generalization of the cylindrical contact homology fromLecture 10. What we saw there was a special case of this where the combinatorial ectures on Symplectic Field Theory factor κ γ /κ u did not appear because we were restricting to a homotopy class in whichall orbits were simply covered, and all holomorphic cylinders were thus somewhereinjective.The presence of the factor κ γ /κ u deserves further comment. According to theabove formula, we have ∂ q γ = X γ ′ ,γ ′′ ,A,A ′ X u ∈M A ( γ,γ ′ ) X v ∈M A ′ ( γ ′ ,γ ′′ ) e A + A ′ κ γ κ γ ′ ǫ ( u ) ǫ ( v ) κ u κ v q γ ′′ , hence ∂ = 0 holds if and only if for all A ∈ H ( M ) and all pairs of good orbits γ + , γ − ,(12.14) X γ X B + C = A X ( u,v ) ∈M B ( γ + ,γ ) ×M C ( γ ,γ − ) κ γ κ u κ v ǫ ( u ) ǫ ( v ) = 0 . If γ + and γ − happen to be simply covered orbits, then u and v in this expressionalways have trivial automorphism groups and it is clear what this sum means: everysuch pair ( u, v ) ∈ M B ( γ + , γ ) ×M C ( γ , γ − ) corresponds to exactly κ γ distinct holo-morphic buildings obtained by different choices of decoration, so (12.14) is the countof boundary points of the compactified 1-dimensional manifold of index 2 cylinders M A ( γ + , γ − ) / R . This sum skips over all bad orbits γ , but this is fine because when-ever the breaking orbit is bad, there are evenly many choices of decoration such thathalf of these choices cancel the other half when counted with the correct signs.To understand why this formula is still correct in the presence of automorphisms,let us outline two equivalent approaches.The easiest option is to instead consider moduli spaces with asymptotic markers,which never have automorphisms: removing unnecessary factors of κ γ + and κ γ − thentransforms (12.14) into X γ X B + C = A κ γ M $ B ( γ + , γ ) · M $ C ( γ , γ − ) = 0 . Now since each pair ( u, v ) ∈ M $ B ( γ + , γ ) ×M $ C ( γ , γ − ) carries a canonical decorationand thus determines a holomorphic building, the division by κ γ accounts for thefact that M $ B ( γ + , γ ) · M $ C ( γ , γ − ) overcounts the set of broken cylinders from γ + to γ − with asymptotic markers at γ ± by precisely this factor, as a simultaneousadjustment of the marker at γ in both u ∈ M $ B ( γ + , γ ) and v ∈ M $ C ( γ , γ − )produces the same decoration and therefore the same building.The following alternative perspective will be more useful when we generalizebeyond cylinders in the next section. We can directly count points in ∂ M A ( γ + , γ − ),though as we saw in § M A ( γ + , γ − ) = M $ A ( γ + , γ − ) /G, where G ∼ = Z κ γ + × Z κ γ − is a finite group acting by adjustment of the asymptoticmarkers. Since M $ A ( γ + , γ − ) is a compact oriented 1-manifold with boundary underAssumption 12.1, the signed count of its boundary points is 0. We can ignore Chris Wendl buildings broken along bad orbits in this count, since these always come in cancellingpairs. Let us now transform this into a count of buildings ( u | Φ | v ) ∈ ∂ M A ( γ + , γ − )broken along good orbits γ : here u ∈ M B ( γ + , γ ) and v ∈ M C ( γ , γ − ) for somehomology classes with B + C = A , and Φ is a decoration which describes how toglue the ends of u and v at γ . The automorphism group of such a building is thesubgroup Aut( u | Φ | v ) ⊂ Aut( u ) × Aut( v )consisting of all pairs ( ϕ, ψ ) ∈ Aut( u ) × Aut( v ) that define the same rotation at thetwo punctures asymptotic to γ ; note that this group does not actually depend onthe decoration Φ. Since we’re talking about cylinders, we can be much more specific:we have Aut( u ) = Z κ u and Aut( v ) = Z κ v , and if both are regarded as subgroups ofU(1), Aut( u | Φ | v ) = Z κ u ∩ Z κ v = Z gcd( κ u ,κ v ) , which is injected into Aut( u ) × Aut( v ) by ψ ( ψ, ψ ). The boundary of M $ A ( γ + , γ − )can be understood likewise as a space of equivalence classes[( u, v )] ∈ (cid:0) M $ B ( γ + , γ ) × M $ C ( γ , γ − ) (cid:1) (cid:14) ∼ , where two such pairs are equivalent if their asymptotic markers at the ends as-ymptotic to γ determine the same decoration. Now observe that the group G ∼ = Z κ γ + × Z κ γ − also acts on buildings in ∂ M $ A ( γ + , γ − ), and the stabilizer of this actionat ( u, v ) is Aut( u | Φ | v ), hence each ( u | Φ | v ) ∈ ∂ M A ( γ + , γ − ) gives rise to | G | gcd( κ u ,κ v ) terms in the count of ∂ M $ A ( γ + , γ − ), implying(12.15) X ( u | Φ | v ) ∈ ∂ M A ( γ + ,γ − ) ǫ ( u ) ǫ ( v )gcd( κ u , κ v ) = 0 . Finally, notice that while each pair ( u, v ) ∈ M B ( γ + , γ ) × M C ( γ , γ − ) determinesbuildings with κ γ distinct choices of decoration, some of these buildings may beequivalent: every pair of automorphisms ( ϕ, ψ ) ∈ Aut( u ) × Aut( v ) transforms abuilding ( u | Φ | v ) by potentially changing the decoration Φ, thus producing an equiv-alent building. This action on buildings is trivial if and only if ( ϕ, ψ ) ∈ Aut( u | Φ | v ),hence every pair ( u, v ) ∈ M B ( γ + , γ ) × M C ( γ , γ − ) gives rise to exactly κ γ (cid:12)(cid:12) (Aut( u ) × Aut( v )) (cid:14) Aut( u | Φ | v ) (cid:12)(cid:12) = κ γ gcd( κ u , κ v ) κ u κ v elements of ∂ M A ( γ + , γ − ), so that (12.15) becomes X γ X B + C = A X ( u,v ) ∈M B ( γ + ,γ ) ×M C ( γ ,γ − ) ǫ ( u ) ǫ ( v )gcd( κ u , κ v ) κ γ gcd( κ u , κ v ) κ u κ v = X γ X B + C = A X ( u,v ) ∈M B ( γ + ,γ ) ×M C ( γ ,γ − ) ǫ ( u ) ǫ ( v ) κ γ κ u κ v = 0 , reproducing (12.14). ectures on Symplectic Field Theory Now let’s try to justify the formula H = 0. The product of H with itself is theformal sum over all pairs of index 1 curves u, v ∈ M σ ( J ) / R of certain monomials:in particular if these two curves respectively have genus g u and g v , homology classes A u and A v , and asymptotic orbits γ ± u and γ ± v , then the corresponding term in H is ǫ ( u ) ǫ ( v ) | Aut σ ( u ) || Aut σ ( v ) | ~ g u + g v − e A u + A v q γ − u p γ + u q γ − v p γ + v . Before we can add up all monomials of this form, we need to put all the q and p variables in the same order: within each of the products q γ − u , p γ + u and so forth this issimply a matter of permuting the variables and changing signs as appropriate, butthe interesting part is the product p γ + u q γ − v , for which we can apply the commutationrelations (12.9) to put all q variables before all p variables. Before discussing howthis works in general, let us consider a more specific example.Assume γ i for i = 1 , n − µ CZ ( γ i ) even, so thecorresponding q and p variables have even degree, and suppose γ + u = ( γ , γ , γ ) , γ − v = ( γ , γ ) . After applying the relation p γ q γ = q γ p γ + κ γ ~ a total of five times, one obtainsthe expansion p γ p γ p γ q γ q γ = q γ p γ p γ + 4 κ γ ~ q γ p γ p γ + 2 κ γ ~ p γ , thus contributing a total of three terms to H , namely the products of the factor ǫ ( u ) ǫ ( v ) | Aut( u ) || Aut( v ) | e A u + A v with each of the expressions ~ g u + g v − q γ − u q γ p γ p γ p γ + v , (12.16) 4 κ γ ~ g u + g v − q γ − u q γ p γ p γ p γ + v , (12.17) 2 κ γ ~ g u + g v q γ − u p γ p γ + v . (12.18)As shown in Figure 12.2, this sum of three terms can be interpreted as the count ofall possible holomrphic buildings obtained by gluing v on top of u together with acollection of trivial cylinders. Indeed, since γ + u and γ − v include two matching orbits(which also happen to be the same one), there are several choices to be made:(1) The top-right picture shows what we might call the “stupid gluing,” inwhich no ends of u are matched with any ends of v , but all are insteadglued to trivial cylinders, thus producing a disconnected building. Thispossibility is encoded by (12.16), and we will see that in the total sumforming H , this term gets cancelled out by a similar term for the stupidgluing of u on top of v .(2) The lower-left picture shows the building obtained by gluing one end of u to an end of v along the matching orbit γ . This option is encodedby (12.17), where the factor 4 κ γ appears because there are precisely 4 κ γ distinct buildings of this type: indeed, there are four choices of which endof u should be glued to which end of v , and for each of these, a further κ γ Chris Wendl uu uu vv vv γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − u γ − u γ − u γ − u γ + v γ + v γ + v γ + v Figure 12.2.
Three possible ways of gluing the curves u and v along with trivial cylinders to form index 2 curves.choices of the decoration. The arithmetic genus of the resulting building is g u + g v , as represented by the factor ~ g u + g v − .(3) The lower-right picture is encoded by (12.18): here there are two choices ofbijections between the two pairs of punctures asymptotic to γ , and takingthe choices of decoration at each breaking orbit into account, we obtain thecombinatorial factor 2 κ γ . The presence of two nontrivial breaking orbitsincreases the arithmetic genus to g u + g v + 1, as encoded in the factor ~ g u + g v .You may now be able to extrapolate from the above example why the commu-tator algebra we’ve defined encodes gluing of holomorphic curves in the symplec-tization and thus leads to the relation H = 0. Think of the algorithm by whichyou change q γ − u p γ + u q γ − v p γ + u into a sum of products with all q ’s appearing before p ’s:for the first q you see appearing after a p , move it past each p for different orbits(changing signs as necessary) until it encounters a p for the same orbit. Now youreplace p γ q γ with ( − | p γ || q γ | q γ p γ + κ γ ~ , turning one product into a sum of two. Thisrepresents a choice between two options: either you move q γ past p γ and apply the ectures on Symplectic Field Theory usual sign change, or you eliminate them both but replace them with the combina-torial factor κ γ and an extra ~ . Then you continue this process until all q ’s appearbefore all p ’s.The key point is that the process of gluing v on top of u in all possible ways isgoverned by exactly the same algorithm : first consider the disjoint union of the twocurves as a single disconnected curve, with its punctures ordered in the same wayin which their orbits appear in the monomial. Now reorder negative punctures of v and positive punctures of u , changing orientations as appropriate, until you see twosuch punctures next to each other approaching the same orbit γ . Here you have twooptions: either glue them together, or don’t glue them but exchange their order. Ifyou exchange the order, then you may again have to change orientations (dependingon the parity of n − µ CZ ( γ )), but if you glue, then you have κ γ distinct choicesof decoration and will also increase the arithmetic genus of the eventual buildingby 1. In this way, every individual term in the final expansion of q γ − u p γ + u q γ − v p γ + u represents a particular choice of which positive of ends of u should or should not beglued to which negative ends of v . Additional factors of ~ appear to keep track ofthe increase in arithmetic genus, and covering multiplicities of the breaking orbitsalso appear due to distinct choices of decorations. At the end these must still bedivided by orders of automorphism groups in order to avoid counting equivalentbuildings separately. Fleshing out these details leads to the following explanationfor the relation H = 0: Proposition . Let ∂ M σ ( J ) denote the space of two-level holomorphicbuildings in M ( J ) that have total index and no bad asymptotic or breaking orbits,divided by the equivalence relation that forgets the order of the punctures. Then H = X u ∈ ∂ M σ ( J ) ǫ ( u ) | Aut σ ( u ) | ~ g − e A q γ − p γ + , where the terms in each monomial are determined by u ∈ ∂ M σ ( J ) as follows:(1) g is the arithmetic genus of u ;(2) A is the equivalence class of [ u ] ∈ H ( M ) in H ( M ) /G ;(3) γ ± = ( γ ± , . . . , γ ± k ± ) are the asymptotic orbits of u after arbitrarily fixingorderings of its positive and negative punctures;(4) ǫ ( u ) ∈ { , − } is the boundary orientation at u determined by the chosencoherent orientations on M $ ( J ) . Specifically, given the chosen ordering ofthe punctures and an arbitrary choice of asymptotic markers at each punc-ture, u determines a boundary point of a -dimensional connected compo-nent of M $ ( J ) , and we define ǫ ( u ) = +1 if and only if the orientation of M $ ( J ) at this point is outward. Once again ǫ ( u ) and q γ − p γ + change signs in the same way under any reorderingof the punctures, so their product is well defined, and there is no dependence onchoices of markers since bad orbits have been excluded. Chris Wendl
Proof of Proposition 12.12.
Our original formula for H gives rise to anexpansion H = X ( u,v ) ∈M σ ( J ) / R ×M σ ( J ) / R ǫ ( u ) ǫ ( v ) | Aut σ ( u ) || Aut σ ( v ) | ~ g u + g v − e A u + A v q γ − u p γ + u q γ − v p γ + v . As explained in the previous paragraph, the process of reordering p γ + u q γ − v to put all q ’s before p ’s produces an expansion, each term of which can be identified with aspecific choice of which positive punctures of u should be glued to which negativepunctures of v . If k punctures are glued, then the resulting power of ~ is g u + g v − k , corresponding to the fact that the resulting building has arithmetic genus g u + g v + k −
1. We claim that the term for k = 0 is cancelled out by the correspondingterm of H that has the roles of u and v reversed. To see this, imagine first the casewhere u and v have no asymptotic orbits in common, hence no nontrivial gluingsare possible and all the q and p variables in the expression supercommute with eachother. Then since both curves have index 1, the monomials q γ − u p γ + u and q γ − v p γ + v mustboth have odd degree, implying q γ − u p γ + u q γ − v p γ + v = − q γ − v p γ + v q γ − u p γ + u and thus the desired cancellation. If u and v do have orbits in common, then theresult for the k = 0 terms is still not any different from this: all signs still changein the same way when applying [ p γ , q γ ] = κ γ ~ to change p γ q γ into q γ p γ , we simplyignore the extra term κ γ ~ since it is only relevant for gluings with k >
0. Thisproves the claim, and consequently, that the expansion resulting from the curves u and v has no term containing ~ g u + g v − .The combinatorial factors can be explained as follows. The commutator expan-sion for p γ + u q γ − v automatically produces combinatorial factors that count the differentpossible gluings, but if u and v have automorphisms, then not all of these give in-equivalent buildings. This part of the discussion is a straightforward extension ofwhat we did for cylindrical contact homology at the end of § σ ( u ) × Aut σ ( v )) (cid:14) Aut σ ( u ) , where for a building u formed by endowing the pair ( u, v ) with decorations, Aut σ ( u )denotes the subgroup consisting of pairs ( ϕ, ψ ) ∈ Aut σ ( u ) × Aut σ ( v ) that preservepairs of breaking punctures along with their decorations. This is what changes thefactor | Aut σ ( u ) || Aut σ ( v ) | into | Aut σ ( u ) | as in the statement of the proposition. (cid:3) The theorem that H = 0 now follows once you believe the propaganda from § P u ∈ ∂ M σ ( J ) ǫ ( u ) | Aut σ ( u ) | is the correct way to count the boundarypoints of M σ ( J ). As we did with cylindrical contact homology, we can use the obvi-ous projection M $ ( J ) → M σ ( J ) to reduce this to the fact that if the 1-dimensionalcomponents of M $ ( J ) are manifolds (which is true if Assumption 12.1 holds), thenthe integer-valued signed count of their boundary points vanishes. ectures on Symplectic Field Theory H ( M ) has torsion? The main consequence for SFT if H ( M )has torsion is that one cannot define an integer grading, though there is always acanonical Z -grading. The setup in § reference curves C , . . . , C r ⊂ M are required to form a basis of H ( M ) / torsion, so for every integral homology class[ γ ], there is a unique collection of integers m , . . . , m r such that [ γ ] = P i m i [ C i ] ∈ H ( M ; Q ). Instead of spanning surfaces for each orbit, one can define spanningchains C γ , which are singular 2-chains with rational coefficients satisfying ∂C γ = X i m i [ C i ] − [ γ ]for the aforementioned set of integers m i ∈ Z . Note that C γ must in generalhave nonintegral coefficients since P i m i [ C i ] and [ γ ] might not be homologous in H ( M ; Z ), so C γ cannot always be represented by a smooth map of a surface. Oneconsequence of this is that the absolute homology class associated to an asymptoti-cally cylindrical holomorphic curve u : ˙Σ → R × M will now be rational,[ u ] ∈ H ( M ; Q ) , and we must therefore take G to be a linear subspace G ⊂ H ( M ; Q ) . Another consequence is that we cannot use capping chains to transfer trivializationsfrom the reference curves to the orbits, so there is no natural way to define µ CZ ( γ )as an integer. The easiest thing to do instead is to take the mod 2 Conley-Zehnderindex µ CZ ( γ ) ∈ Z and define all degrees of generators as either even or odd with no further distinction.In particular, we now have | q γ | = n − µ CZ ( γ ) ∈ Z , | p γ | = n − − µ CZ ( γ ) ∈ Z , while ~ and all elements of R = Q [ H ( M ; Q ) /G ] are even. With these modifications,the rest of the discussion also becomes valid for the case where H ( M ) has torsion,and leads to Z -graded contact invariants. The combinatorial factors appearing inour definition of H may at first look slightly different from what appears elsewhere inthe literature. Actually, most papers seem to agree on this detail, but various subtledifferences and ambiguities in notation mean that it sometimes requires intenseconcentration to recognize this fact.The original propaganda paper [ EGH00 ] expresses everything in terms of modulispaces with asymptotic markers, and the formula for H in § In fact there is a bit more than a Z -grading, see [ EGH00 , § Chris Wendl is expressed in a slightly more general form involving marked points) agrees withour (12.8).Cieliebak and Latschev [
CL09 , §
2] write down the same formula in terms of mod-uli spaces that have no asymptotic markers but remember the order of the punctures,thus it includes some factorials that do not appear in (12.6) but is missing the κ γ terms of (12.8). The notation n g (Γ − , Γ + ) used in [ CL09 ] for curve counts must beunderstood implicitly to include rational weights arising from automorphisms (ormultivalued perturbations, as the case may be).My paper with Latschev [
LW11 ] uses moduli spaces with asymptotic markersand attempts to write down the same formula as in [
EGH00 , CL09 ], but gets itslightly wrong due to some missing κ γ terms that should appear in front of each ∂∂q γ .Mea culpa.For cylindrical contact homology, the combinatorial factors in § Bou03 ]. As observed by Nelson [
Nel13 , Remark 8.3], thereare other conventions for ∂ CCH that appear in the literature and lead to equivalenttheories: in particular it is possible to replace (12.13) with ∂ CCH q γ := X γ ′ ,A κ γ ′ X u ∈M A ( γ,γ ′ ) ǫ ( u ) κ u e A q γ ′ . One can derive this from the same definition of H by applying a “change of co-ordinates” to the algebra A [[ ~ ]], or equivalently, by choosing a slightly differentrepresentation of the operator algebra defined by the p γ and q γ variables. To avoidconfusion, let us write the generators of A as x γ instead of q γ , and then define theoperators q γ and p γ on A [[ ~ ]] by q γ = κ γ x γ , p γ = ~ ∂∂x γ . These operators still satisfy [ p γ , q γ ] = κ γ ~ and thus define an equivalent theory, butthe resulting differential operator D SFT on A [[ ~ ]] now includes extra factors of κ γ for the negative punctures instead of the positive punctures. Q , Z or Z ? While we were able to use Z coefficientsfor cylindrical contact homology in a primitive homotopy class in Lecture 10, a quickglance at any version of the formula for H should make the reader very skepticalabout doing this for more general versions of SFT. The existence of curves withautomorphisms means that H always contains terms with rational (but nonintegral)coefficients. And this is only what is true in the fictional world of Assumption 12.1:in the general version of the theory, we expect to have to replace expressions like P u ǫ ( u ) | Aut( u ) | with counts of 0-dimensional weighted branched orbifolds with rationalweights, arising as zero-sets of generic multisections. In this case we not only obtainrational counts but may also lose all control over the sizes of the denominators.A similar phenomenon occurs in general versions of Gromov-Witten theory. Forinstance, in the approach of Cieliebak-Mohnke [ CM07 ] for the rational Gromov-Witten invariants of a closed symplectic manifold ( W n , ω ) with [ ω ] ∈ H ( W ; Q ),the invariants are defined by replacing the usual moduli space M ,m ( J, A ) by a space ectures on Symplectic Field Theory M ,m + N ( J, A ; Y ) consisting of J -holomorphic spheres u : S → W with some largenumber of auxiliary marked points ζ , . . . , ζ N required to satisfy the condition u ( ζ i ) ∈ Y, i = 1 , . . . , N.
Here Y n − ⊂ W n is a J -holomorphic hypersurface with [ Y ] = D · PD([ ω ]) ∈ H n − ( W ) for some degree D ∈ N , and the number of extra marked points is deter-mined by N = A · [ Y ] = D h [ ω ] , A i , so positivity of intersections implies that u only intersects Y at the auxiliary markedpoints. These auxiliary points are convenient for technical reasons involving trans-versality—their role is vaguely analogous to the way that asymptotic markers getrid of isotropy in SFT—but they are not geometrically meaningful, as we’d actuallyprefer to count curves in M ,m ( J, A ). Every such curve has N intersections with Y ,so accounting for permutations, it lifts to N ! distinct elements of M ,m + N ( J, A ; Y ),and the correct count is therefore obtained as an integer count of curves in the latterspace divided by N !. Perturbing to achieve transversality breaks the symmetry,however, so there is no guarantee that counting curves in M ,m + N ( J, A ; Y ) willproduce a multiple of N !, and moreover, N could be arbitrarily large since oneneeds to take hypersurfaces of arbitrarily large degree in order to show that theinvariants don’t depend on this choice. For these reasons, the resulting Gromov-Witten invariants are rational numbers rather than integers, and their denominatorscannot be predicted or bounded.The upshot of this discussion is that there is probably no hope of defining SFTwith integer coefficients in general, much less with Z coefficients—for this reasonthe inclusion of orientations in the picture is unavoidable. That is the bad news.The good news however is that whenever formulas like P u ǫ ( u ) | Aut( u ) | can be takenliterally as a count of curves, the chain complex ( A [[ ~ ]] , D SFT ) can in fact be definedwith Z coefficients, and one can even reduce to a Z version in order to ignore signs.A special case of this was observed for cylindrical contact homology in [ Nel15 , Re-mark 1.5], and you may notice it already when you look at the formula (12.13) for ∂ CCH : the factor κ γ /κ u is always an integer since the multiplicity of a holomorphiccylinder always divides the covering multiplicity of both its asymptotic orbits. Sur-prisingly, something similar turns out to be true for the much larger chain complexof SFT. The following result is stated under Assumption 12.1 for safety’s sake, but inlight of the discussion in § Proposition . If Assumption 12.1 holds then the rational coefficients n g ( γ , γ ′ , k ) in the formula (12.11) for D SFT q γ are all integers. Corollary . Under Assumption 12.1, there exist well-defined chain com-plexes ( A Z [[ ~ ]] , D SFT ) and ( A Z [[ ~ ]] , D SFT ) , where for a general commutative ring R , A R denotes the graded supercommutativeunital algebra over R [ H ( M ) /G ] generated by the q γ variables for good Reeb orbits γ .The differentials D SFT on A Z [[ ~ ]] and A Z [[ ~ ]] are defined by the same formula ason A [[ ~ ]] , where in the Z case we are free to set all signs ǫ ( u ) equal to . Chris Wendl
Proof of Proposition 12.13.
We need to show that expressions of the form κ γ + | Aut σ ( u ) | ∂∂q γ +1 . . . ∂∂q γ + k + q γ produce integer coefficients for every holomorphic curve u with asymptotic orbits γ ± = ( γ ± , . . . , γ ± k ± ) and every tuple γ = ( γ , . . . , γ m ). It suffices to consider thespecial case γ = γ + , as the derivative in question is only nonzero on monomialsthat are divisible by q γ + . Up to a sign change, we can reorder the orbits and write γ + in the form γ + = ( γ , . . . , γ | {z } m , . . . , γ N , . . . , γ N | {z } m N )for some finite set of distinct orbits γ , . . . , γ N and numbers m i ∈ N , i = 1 , . . . , N .We then have κ γ + | Aut σ ( u ) | ∂∂q γ +1 . . . ∂∂q γ + k + q γ + = κ m γ . . . κ m N γ N | Aut σ ( u ) | (cid:18) ∂∂q γ (cid:19) m . . . (cid:18) ∂∂q γ N (cid:19) m N (cid:0) q m γ . . . q m N γ N (cid:1) = ± κ m γ . . . κ m N γ N m ! . . . m N ! | Aut σ ( u ) | . (12.19)We claim that this number is always an integer. Indeed, if Aut σ ( u ) is nontrivial, then u : ˙Σ → R × M is a multiple cover u = v ◦ ϕ for some holomorphic branched cover ϕ : (Σ , j ) → (Σ ′ , j ′ ) and somewhere injective curve v : ( ˙Σ ′ = Σ ′ \ Γ ′ , j ′ ) → ( R × M, J ).Automorphisms ψ ∈ Aut σ ( u ) thus define biholomorphic maps on (Σ , j ) that permuteeach of the sets of punctures asymptotic to the same orbit. Given any puncture z ∈ Γ where u is asymptotic to γ i , the Aut σ ( u )-orbit of z consists of ℓ ≤ m i otherpunctures also asymptotic to γ i , and its stabilizer is a cyclic subgroup of order k = | Aut σ ( u ) | /ℓ , acting on a neighborhood of z by biholomorphic rotations. Itfollows that κ γ i is divisible by k , hence κ γ i ℓ | Aut σ ( u ) | ∈ N , and (12.19) is a multiple of this. (cid:3) Remark . Since 1 = − A Z , anticommuting elements of A Z [[ ~ ]] ac-tually commute, so unless one imposes extra algebraic conditions in the case of Z coefficients, higher powers of odd generators p γ and q γ do not vanish. Nonetheless,these powers still do not appear in H , so the complex ( A Z [[ ~ ]] , D SFT ) ignores curveswith multiple ends approaching an orbit of odd degree (and also bad orbits, for thatmatter).ECTURE 13
Contact invariantsContents BV ∞ -algebra 288 In the previous lecture, we introduced an operator algebra defined via the su-percommutators [ p γ , q γ ] = κ γ ~ , then we defined the SFT generating function H = X u ∈M σ ( J ) / R ǫ ( u ) | Aut σ ( u ) | ~ g − e A q γ − p γ + and proved (modulo transversality) that H = 0. The generating function is a formalpower series whose coefficients are rational counts of holomorphic curves, and thesecounts are strongly dependent on the choices of contact form α , almost complexstructure J ∈ J ( α ) and further auxiliary data such as coherent orientations. Thusin contrast to Gromov-Witten theory, the generating function does not define aninvariant, but one can follow the standard prescription of Floer-type theories anddefine invariants via homology. We saw that for the natural representation A [[ ~ ]] ofthe operator algebra defined by setting p γ = κ γ ~ ∂∂q γ , H defines a differential operator D SFT : A [[ ~ ]] → A [[ ~ ]] with D = 0. One of our goals in this lecture will be toexplain (again modulo transversality) why the resulting homology H SFT ∗ ( M, ξ ; R ) = H ∗ ( A [[ ~ ]] , D SFT )is an invariant of the contact structure. We will then use it to define simpler nu-merical invariants that detect symplectic fillability properties of contact manifolds.But first, A [[ ~ ]] is not the only possible representation of the operator algebraof SFT: other choices lead to different invariants with different algebraic structures.Let’s begin by describing the original hierarchy of contact invariants that were out-lined in [ EGH00 ]. Chris Wendl
Remark . Throughout this lecture, we assume for simplicity that H ( M )has no torsion, and the same assumption is made about cobordisms in § H ( M ) with H ( M ; Q ) and assume always that the grading is Z ; see § In the following, (
M, ξ ) is a (2 n − α and almost complex structure J ∈ J ( α ) for which the optimistictransversality condition (Assumption 12.1) of Lecture 12 is assumed to hold. Wefix also the auxiliary data described in § G ⊂ H ( M )which determines the coefficient ring R = Q [ H ( M ) /G ] . Each of the differential graded algebras described below then carries the same grad-ing that was described in that lecture, i.e. there is always at least a Z -grading, andit lifts to Z if H ( M ) is torsion free and c ( ξ ) | G = 0, or possibly Z N if N ∈ N is thesmallest possible value for c ( A ) with A ∈ G . We start with some seeminglytrivial algebraic observations. First, the relation H = 0 is equivalent to[ H , H ] = 0 . Remember that [ , ] is a super -commutator, so [ F , F ] = 0 holds automatically foroperators F with even degree, but H is odd, and for odd operators the commutatoris defined by [ F , G ] = FG + GF , hence [ H , H ] = 2 H . Formally speaking [ , ] is a super Lie bracket and thus satisfies the “super Jacobi identity”:(13.1) (cid:2) F , [ G , K ] (cid:3) + ( − | F || G | + | F || K | (cid:2) G , [ K , F ] (cid:3) + ( − | F || K | + | G || K | (cid:2) K , [ F , G ] (cid:3) = 0 . A consequence of this is that in order to create a homology theory out of H , we don’tabsolutely need to find a representation of the entire operator algebra: it sufficesto find a representation of the induced super Lie algebra. Indeed, suppose V is agraded R [[ ~ ]]-module and L is a linear grading-preserving map that associates tooperators F (expressed as power series functions of p ’s, q ’s and ~ with coefficientsin R ) an R [[ ~ ]]-linear map L F : V → V such that L [ F , G ] = L F L G − ( − | F || G | L G L F for every pair of operators F , G . Then the R [[ ~ ]]-linear map L H : V → V satisfies L H = 12 [ L H , L H ] = 12 L [ H , H ] = 0 , hence ( V, L H ) is a chain complex. The complex ( A [[ ~ ]] , D SFT ) was a special case ofthis, in which we represented the super Lie algebra via a faithful representation ofthe whole operator algebra.
Exercise . Verify (13.1). ectures on Symplectic Field Theory
Remark . To see where the signs in (13.1)come from, it suffices to know the following basic rule of superalgebra: for anypair of Z -graded vector spaces V and W , the natural “commutation” isomorphism c : V ⊗ W → W ⊗ V is defined on homogeneous elements by c ( v ⊗ w ) = ( − | v || w | w ⊗ v. For any permutation of a finite tuple of Z -graded vector spaces, one can derive theappropriate isomorphism from this: in particular the cyclic permutation isomor-phism σ : X ⊗ Y ⊗ Z → Y ⊗ Z ⊗ X takes the form σ = ( ⊗ c ) ◦ ( c ⊗ ) : x ⊗ y ⊗ z ( − | x || y | + | x || z | y ⊗ z ⊗ x. Writing the Jacobi identity as [ · , [ · , · ]] ◦ ( + σ + σ ) = 0 then produces (13.1). Inthis sense, it only differs from the usual Jacobi identity in being based on a differentdefinition of the commutation isomorphism V ⊗ W → W ⊗ V . For more on thisperspective, see [ Var04 , § W denote the graded unital algebra consisting of formal power series X γ ,k f γ ,k ( q ) ~ k p γ , where the sum ranges over all integers k ≥ γ = ( γ , . . . , γ m )of good Reeb orbits for m ≥
0, and the f γ ,k are polynomial functions of the q γ variables, with coefficients in R . Note that the case of the empty set of orbits isincluded here, which means p γ = 1. The multiplicative structure of W is definedvia the usual (super)commutation relations, and its elements can be interpreted asoperators. If we now associate to each F ∈ W the R [[ ~ ]]-linear map D F : W → W : G [ F , G ] , then the Jacobi identity (13.1) implies D [ F , G ] = D F D G − ( − | F || G | D G D F . This is just the graded version of the standard adjoint representation of a Lie algebra.The only problem in applying this idea to define a differential(13.2) D H : W → W : F [ H , F ]is that H is not technically an element of W : indeed, H contains terms of order − ~ , thus H ∈ ~ W . On the other hand, the failure of supercommutativity in W is a “phenomenon oforder ~ ,” i.e. since every nontrivial commutator contains a factor of ~ , we have[ F , G ] = O ( ~ ) for all F , G ∈ W . Here and in the following we use the symbol O ( ~ k ) Chris Wendl to denote any element of the form ~ k F for F ∈ W . As a consequence, [ H , F ] ∈ W whenever F ∈ W , hence (13.2) is well defined, and the Jacobi identity now implies D H = 0 . The homology of the resulting chain complex gives another version of what is oftencalled full SFT , H W ∗ ( M, ξ ; R ) := H ∗ ( W , D H ) . A proof (modulo transversality) that this defines a contact invariant is outlined in[
EGH00 , § H SFT ∗ ( M, ξ ; R ),so I will skip it since I don’t have any applications of H W ∗ ( M, ξ ; R ) in mind. As faras I am aware, no contact topological applications of this invariant or computationsof it (outside the trivial case—see § H W ∗ ( M, ξ ; R ) actually has much more algebraic structure than H SFT ∗ ( M, ξ ; R ). Indeed, using the identities[ F , GK ] = [ F , G ] K + ( − | F || G | G [ F , K ] , [ FG , K ] = F [ G , K ] + ( − | G || K | [ F , K ] G , (13.3)one sees that D H : W → W satisfies a graded Leibniz rule, D H ( FG ) = ( D H F ) G + ( − | F | F D H G . It follows that D H : W → W is also a derivation with respect to the bracket structureon W , i.e. D H [ F , G ] = [ D H F , G ] + ( − | F | [ F , D H G ]for all F , G ∈ W . As a consequence, the product and bracket structures on W descend to H W ∗ ( M, ξ ; R ), giving it the structure of a Weyl superalgebra .As a matter of interest, we observe that ( W , D H ), as with ( A [[ ~ ]] , D SFT ) in theprevious lecture, can be defined with Z or Z coefficients whenever the transversalityresults are good enough to take the usual expression P u ǫ ( u ) | Aut σ ( u ) | literally as a countof holomorphic curves. This result is of limited interest since it cannot hold in gen-eral cases where transversality for multiple covers is impossible without multivaluedperturbations—nonetheless I find it amusing. Proposition . If Assumption 12.1 in Lecture 12 holds, then D H is also welldefined if the ring R = Q [ H ( M ) /G ] is replaced by Z [ H ( M ) /G ] or Z [ H ( M ) /G ] . Proof.
Since D H is a derivation, it suffices to check that for every good Reeborbit γ , D H q γ and D H p γ are each sums of monomials of the form ce A ~ k q γ − p γ + with The same arguments used to define SFT chain complexes over the integers can also be appliedto the chain maps involved in the proof of invariance (see § should be defined over the integers if transversality can be achieved for multiple covers. There are knownsituations however in which this cannot hold: even if the chain complexes are well defined over Z ,invariance may hold only over Q , due to the failure of transversality in cobordisms. See [ Hut ]. ectures on Symplectic Field Theory coefficients c ∈ Z . Suppose u ∈ M ( J ) is an index 1 holomorphic curve with positiveand/or negative asymptotic orbits γ ± = ( γ ± , . . . , γ ± | {z } m ± , . . . , γ ± k ± , . . . , γ ± k ± | {z } m ± k ± ) , where γ ± i = γ ± j for i = j . We can assume all the orbits γ ± i are good and that m ± i = 1whenever n − µ CZ ( γ ± i ) is odd. Up to a sign and factors of e A and ~ which arenot relevant to this discussion, u then contributes a monomial H u := 1 | Aut σ ( u ) | q m − γ − . . . q m − k − γ − k − p m +1 γ +1 . . . p m + k + γ + k + to H . The commutator [ H u , q γ ] vanishes unless γ is one of the orbits γ +1 , . . . , γ + k + ,so suppose γ = γ + k + . If n − µ CZ ( γ ) is odd, then m := m + k + = 1, and (13.3) with[ p γ , q γ ] = κ γ ~ implies[ H u , q γ ] = 1 | Aut σ ( u ) | (cid:20) q m − γ − . . . q m − k − γ − k − p m +1 γ +1 . . . p m + k + − γ + k + − p γ , q γ (cid:21) = κ γ | Aut σ ( u ) | ~ q m − γ − . . . q m − k − γ − k − p m +1 γ +1 . . . p m + k + − γ + k + − . The fraction in front of this expression is an integer since u can have only oneend asymptotic to γ , and κ γ is thus divisible by the covering multiplicity of u . If n − µ CZ ( γ ) is even, then we generalize this calculation by using (13.3) to write[ p mγ , q γ ] = mκ γ ~ p m − γ , so then, [ H u , q γ ] = 1 | Aut σ ( u ) | (cid:20) q m − γ − . . . q m − k − γ − k − p m +1 γ +1 . . . p m + k + − γ + k + − p mγ , q γ (cid:21) = κ γ m | Aut σ ( u ) | ~ q m − γ − . . . q m − k − γ − k − p m +1 γ +1 . . . p m + k + − γ + k + − p m − γ . To see that κ γ m | Aut σ ( u ) | is always an integer, recall from our proof of Prop. 12.13 inthe previous lecture that transformations in Aut σ ( u ) permute each of the sets ofpunctures that are asymptotic to the same Reeb orbit. Suppose the set of positivepunctures of u asymptotic to γ is partitioned by the Aut σ ( u )-action into N subsets,each consisting of ℓ , . . . , ℓ N punctures, where ℓ + . . . + ℓ N = m . If z is a puncturein the i th of these subsets, then its stabilizer is a cyclic subgroup of order k i actingon a neighborhood of z by biholomorphic rotations, where k i ℓ i = | Aut σ ( u ) | . Eachof these orders k i necessarily divides the multiplicity κ γ , so we can write k i a i = κ γ for some a i ∈ N . Putting all this together, we have κ γ m = N X i =1 κ γ ℓ i = N X i =1 k i a i ℓ i = | Aut σ ( u ) | N X i =1 a i . Following this same procedure, you should now be able to verify on your ownthat the coefficient appearing in [ H u , p γ ] is also always an integer. The existence Chris Wendl of a chain complex with Z coefficients follows from this simply by projecting Z to Z . (cid:3) The idea of rational sym-plectic field theory (RSFT) is to extract as much information as possible from genuszero holomorphic curves but ignore curves of higher genus. The algebra of SFTprovides a fairly obvious mechanism for this: RSFT should be what SFT becomesin the “limit as ~ → P := W (cid:14) ~ W , so P is a graded unital algebra generated by the p γ and q γ variables and the co-efficient ring R , but it does not include ~ as a generator. Since all commutatorsin W are in ~ W , the product structure of P is supercommutative. Let us use thedistinction between capital and lowercase letters to denote the quotient projection W → P : F f . We will make an exception for the letter “H”: recall that H is not an element of W since its genus zero terms have order − ~ , but ~ H ∈ W , so we will define h = X u ǫ ( u ) | Aut σ ( u ) | e A q γ − p γ + ∈ P to be the image of ~ H under the projection. The sum in this expression rangesover all R -equivalence classes of index 1 curves with genus zero, so h will serve asthe generating function of RSFT. To encode gluing of genus zero terms, note firstthat the commutator operation would not be appropriate since it prodcues termsfor every possible gluing of two curves, including those which glue genus zero curvesalong more than one breaking orbit to produce buildings with positive arithmeticgenus. We need instead to have an algebraic operation on P that encodes gluingalong only one breaking orbit at a time.You already know what to expect if you’ve ever taken a quantum mechanicscourse: in the “classical limit,” commutators become Poisson brackets. To expressthis properly, we need to make a distinction between differential operators operatingfrom the left or the right: let −−→ ∂∂q γ : W → W denote the usual operator ∂∂q γ , which was previously defined on A [[ ~ ]] but has anobvious extension to W such that −→ ∂∂q γ p γ ′ = 0 for all γ ′ . This operator satisfies thegraded Leibniz rule −−→ ∂∂q γ ( FG ) = −−→ ∂∂q γ F ! G + ( − | q γ || F | F −−→ ∂∂q γ G ! . The related operator ←−− ∂∂q γ : W → W : F F ←−− ∂∂q γ ectures on Symplectic Field Theory is defined exactly the same way on individual variables p γ and q γ , but satisfies aslightly different Leibniz rule,( FG ) ←−− ∂∂q γ = F G ←−− ∂∂q γ ! + ( − | q γ || G | F ←−− ∂∂q γ ! G . The point of writing ←− ∂∂q γ so that it acts from the right is to obey the usual conventionsof superalgebra: signs change whenever the order of two odd elements (or operators)is interchanged. Partial derivatives with respect to p γ can be defined analogouslyon W . With this notation in hand, the graded Poisson bracket on W is definedby(13.4) { F , G } = X γ κ γ F ←−− ∂∂p γ −−→ ∂∂q γ G − ( − | F || G | G ←−− ∂∂p γ −−→ ∂∂q γ F ! , where the sum ranges over all good Reeb orbits. In the same manner, the differentialoperators and the bracket { , } can also be defined on P .It is easy to check that { , } on W almost satisfies a version of (13.3): we have { F , GK } = { F , G } K + ( − | F || G | G { F , K } + O ( ~ ) , { FG , K } = F { G , K } + ( − | G || K | { F , K } G + O ( ~ )(13.5)for all F , G , K ∈ W . The extra terms denoted by O ( ~ ) arise from the fact thatin proving (13.5), we must sometimes reorder products FG by writing them as( − | F || G | GF + [ F , G ], where [ F , G ] = O ( ~ ). Since the terms with ~ disappear in P , the relations become exact in P : { f , gk } = { f , g } k + ( − | f || g | g { f , k } , { fg , k } = f { g , k } + ( − | g || k | { f , k } g (13.6)for all f , g , k ∈ P . Proposition . For all F , G ∈ W , [ F , G ] = ~ { f , g } + O ( ~ ) , and { , } satisfies the conditions of a super Lie bracket on P . Remark . In formulas like the one in the above proposition, we interpret { f , g } ∈ P as an element of W via any choice of R -linear inclusion P ֒ → W that actsas the identity on the generators p γ , q γ . There is ambiguity in this choice due to thenoncommutativity of W , but the ambiguity is in ~ W and thus makes no differenceto the formula. Proof of Proposition 13.5.
The formula is easily checked when F and G are individual variables of the form p γ or q γ ; in fact the extra term O ( ~ ) can beomitted in these cases. The case where F and G are general monomials follows fromthis via (13.3) and (13.5) using induction on the number of variables in the product.This implies the general case via bilinearity. Chris Wendl
Given the formula, the condition { f , g } + ( − | f || g | { g , f } = 0 and the Poissonversion of the super Jacobi identity (13.1) follow from the corresponding propertiesof [ , ]. (cid:3) The proposition implies that our genus zero generating function h ∈ P satisfies0 = ~ [ H , H ] = [ ~ H , ~ H ] = ~ { h , h } + O ( ~ ), thus { h , h } = 0 . This relation can be interpreted as the count of boundary points of all 1-dimensionalmoduli spaces of genus zero curves: indeed, any pair of genus two curves u, v ∈M σ ( J ) / R constributes to { h , h } a term of the form X γ κ γ | Aut σ ( u ) || Aut σ ( v ) | e A u + A v q γ − u p γ + u ←−− ∂∂p γ ! −−→ ∂∂q γ q γ − v ! p γ + v , plus a corresponding term with the roles of u and v reversed. This sums all the mono-mials that one can construct by cancelling one p γ variable from u with a matching q γ variable from v , in other words, constructing a building by gluing v on top of u along one matching Reeb orbit.The graded Jacobi identity will again imply that any representation of the superLie algebra ( P , { , } ) gives rise to a chain complex with h as its differential. Forexample we can take the adjoint representation, P → End R ( P ) : f d f , d f g := { f , g } , which satisfies d { f , g } = d f d g − ( − | f || g | d g d f due to the Jacobi identity. Then d h = 0since h has odd degree and { h , h } = 0, and the homology of rational SFT isdefined as H RSFT ∗ ( M, ξ ; R ) := H ∗ ( P , d h ) . We again refer to [
EGH00 ] for an argument that H RSFT ∗ ( M, ξ ; R ) is an invariantof the contact structure. Notice that Proposition 13.5 yields a simple relationshipbetween the chain complexes ( W , D H ) and ( P , d h ), namely(13.7) D H F = d h f + O ( ~ ) , where d h f is interpreted as an element of W via Remark 13.6. In other words, theprojection W → P : F → f is a chain map. Moreover, d H is a derivation on P withrespect to both the product and the Poisson bracket: this follows via Proposition 13.5and (13.7) from the fact that D H satisfies the corresponding properties on W . Weconclude that H RSFT ∗ ( M, ξ ; R ) inherits the structure of a Poisson superalgebra, andthe map H W ∗ ( M, ξ ; R ) → H RSFT ∗ ( M, ξ ; R )induced by the chain map ( W , D H ) → ( P , d h ) is both an algebra homomorphismand a homomorphism of graded super Lie algebras. ectures on Symplectic Field Theory Contact homology is the most pop-ular tool in the SFT package and was probably the first to be understood beyond themore straightforward cylindrical theory. In situations where cylindrical contact ho-mology cannot be defined due to bubbling of holomorphic planes, the next simplestthing one can do is to define a theory that counts genus zero curves with one positiveend but arbitrary numbers of negative ends (cf. Exercise 10.14 in Lecture 10).The proper algebraic setting for such a theory turns out to be the algebra A generated by the q γ variables, and it can be derived from RSFT by setting all p γ variables to zero. Using the obvious inclusion A ֒ → P , define ∂ CH : A → A by ∂ CH f = d h f | p =0 . We can thus write d h f = ∂ CH f + O ( p ), where O ( p k )will be used generally to denote any formal sum consisting exclusively of terms ofthe form p γ . . . p γ k f for f ∈ P . Now observe that for any good orbit γ , d h p γ = { h , p γ } = − ( − | p γ | X γ ′ p γ ←−− ∂∂p γ ′ ! −−→ ∂∂q γ ′ h ! = − ( − | p γ | ∂ h ∂q γ = O ( p )since every term in h has at least one p variable. It follows that d h ( O ( p )) = O ( p ),so the fact that d h = 0 implies ∂ = 0, and contact homology is defined as HC ∗ ( M, ξ ; R ) := H ∗ ( A , ∂ CH ) . Since d h is a derivation on P , the formula d h f = ∂ CH f + O ( p ) implies that ∂ CH islikewise a derivation on A , so HC ∗ ( M, ξ ; R ) has the structure of a graded super-commutative algebra with unit. Moreover, the projection P → A : f f | p =0 is achain map, giving rise to an algebra homomorphism H RSFT ∗ ( M, ξ ; R ) → HC ∗ ( M, ξ ; R ) . The invariance of HC ∗ ( M, ξ ; R ) will follow from the invariance of H SFT ∗ ( M, ξ ; R ), tobe discussed in § ∂ CH , we can separate the part of h that is linear in p variables,writing h = X γ h γ ( q ) p γ + O ( p ) , where for each good Reeb orbit γ , h γ ( q ) denotes a polynomial in q variables withcoefficients in R . Since elements f ∈ A have no dependence on p variables, we thenhave d h f = { h , f } = X γ κ γ h ←−− ∂∂p γ ! −−→ ∂∂q γ f ! = X γ κ γ h γ ∂ f ∂q γ + O ( p ) , hence ∂ CH f = X γ κ γ h γ ∂ f ∂q γ . Chris Wendl
In particular, ∂ CH acts on each generator q γ ∈ A as ∂ CH q γ = κ γ h γ = X u ǫ ( u ) κ γ Aut σ ( u ) e A q γ − , where the sum is over all R -equivalence classes of index 1 J -holomorphic curves u with genus zero, one positive end at γ , and negative ends γ − , and homology class A ∈ H ( M ) /G . Even the simplest of the three differen-tial graded algebras described above is too large to compute in most cases. Themajor exception is the case of overtwisted contact manifolds.
Theorem . If ( M, ξ ) is overtwisted, then HC ∗ ( M, ξ ; R ) = 0 for all choicesof the coefficient ring R . Remark . If X is an algebra with unit, then saying X = 0 is equivalent tosaying that 1 = 0 in X .The notion of overtwisted contact structures in dimension three was introducedby Eliashberg in [ Eli89 ], who proved that they are flexible in the sense that theirclassification up to isotopy reduces to the purely obstruction-theoretic classificationof almost contact structures up to homotopy. This means in effect that an over-twisted contact structure carries no distinctly contact geometric information, so itshould not be surprising when “interesting” contact invariants such as HC ∗ ( M, ξ )vanish. The three-dimensional case of Theorem 13.7 seems to have been among theearliest insights about SFT: its first appearance in the literature was in [
Eli98 ], anda proof later appeared in a paper by Mei-Lin Yau [
Yau06 ], which includes a briefappendix sketching Eliashberg’s original proof. We will discuss Eliashberg’s proofin detail in Lecture 16.The definitive higher-dimensional notion of overtwistedness was introduced a fewyears ago by Borman-Eliashberg-Murphy [
BEM15 ], following earlier steps in thisdirection by Niederkr¨uger [
Nie06 ] and others. There are now two known proofs ofTheorem 13.7 in higher dimensions: the first uses the fact that since overtwistedcontact manifolds are flexible, they always admit an embedding of a plastikstufe ,which implies vanishing of contact homology by an unpublished result of Bourgeoisand Niederkr¨uger (see [
Bou09 , Theorem 4.10] for a sketch of the argument). Thesecond argument appeals to an even more recent result of Casals-Murphy-Presas[
CMP ] showing that (
M, ξ ) is overtwisted if and only if it is supported by a nega-tively stabilized open book, in which case HC ∗ ( M, ξ ) = 0 was proven by Bourgeoisand van Koert [
BvK10 ].It is not known whether the vanishing of contact homology characterizes over-twistedness, i.e. there are not yet any examples of tight contact manifolds with HC ∗ ( M, ξ ) = 0. I will go out on a limb and say that such examples seem unlikelyto exist in dimension three but are much more likely in higher dimensions; in factvarious candidates are known [
MNW13 , CDvK ], but we do not yet have adequatemethods to prove that any of them are tight. The analogous question about Legen-drian submanifolds and relative contact homology was recently answered by Ekholm ectures on Symplectic Field Theory [ Ekh ], giving examples of Legendrians that are not loose in the sense of Murphy[
Mur ] but have vanishing Legendrian contact homology.Nevertheless, the lack of known counterexamples has given rise to the followingdefinition.
Definition . A closed contact manifold (
M, ξ ) is algebraically over-twisted if HC ∗ ( M, ξ ; R ) = 0 for every choice of the coefficient ring R . Remark . The coefficient ring is not always mentioned in statements of theabove definition, but it should be. We will see in § G ⊂ G ′ ⊂ H ( M ), the natural projection H ( M ) /G ′ → H ( M ) /G induces an algebra homomorphism HC ∗ ( M, ξ ; Q [ H ( M ) /G ′ ]) → HC ∗ ( M, ξ ; Q [ H ( M ) /G ]) . Since algebra homomorphisms necessarily map 1
0, the target ofthis map must vanish whenever its domain does, so for checking Definition 13.9, itsuffices to check the case R = Q [ H ( M )].We’ve seen above that there exist algebra homomorphisms(13.8) H W ∗ ( M, ξ ; R ) → H RSFT ∗ ( M, ξ ; R ) → HC ∗ ( M, ξ ; R ) , thus the vanishing of either of the algebras H W ∗ ( M, ξ ; R ) or H RSFT ∗ ( M, ξ ; R ) withall coefficient rings R is another sufficient condition for algebraic overtwistedness.Bourgeois and Niederkr¨uger observed that, in fact, these conditions are also neces-sary: Theorem
BN10 ]) . For any coefficient ring R , the following conditionsare equivalent:(1) HC ∗ ( M, ξ ; R ) = 0 ,(2) H RSFT ∗ ( M, ξ ; R ) = 0 ,(3) H SFT ∗ ( M, ξ ; R ) = 0 . Proof.
The implications (3) ⇒ (2) ⇒ (1) are immediate from the algebrahomomorphisms (13.8), thus it will suffice to prove (1) ⇒ (3). Suppose 1 = 0 ∈ HC ∗ ( M, ξ ; R ), which means ∂ CH f = 1 for some f ∈ A . Using the obvious inclusion A ֒ → W , this means D H f = 1 − G , where G = O ( p, ~ ), i.e. G is a sum of terms that all contain at least one p γ variableor a power of ~ . It follows that G k = O ( p k , ~ k ) for all k ∈ N , and the infinite sum ∞ X k =0 G k is therefore an element of W , as the coefficient in front of any fixed monomial ~ k p γ inthis sum is a polynomial function of the q variables. This sum is then a multiplicativeinverse of 1 − G , and since 0 = D H f = 0 = − D H G , Chris Wendl it also satisfies D H ((1 − G ) − ) = 0. Using the fact that D H is a derivation, wetherefore have D H (cid:0) (1 − G ) − f (cid:1) = (1 − G ) − (1 − G ) = 1 , implying 1 = 0 ∈ H SFT ∗ ( M, ξ ; R ). (cid:3) All invariance proofs in SFT are based on a generating function analogous to H that counts index 0 holomorphic curves in symplectic cobordisms. The basicdefinition is a straightforward extension of what we saw in Lecture 12, but thereis an added wrinkle due to the fact that, in general, one must include disconnected curves in the count. First some remarks aboutthe category we are working in. Since the stated purpose of SFT is to define invari-ants of contact structures, we have been working since Lecture 12 with symplectiza-tions of contact manifolds rather than more general stable Hamiltonian structures.We’ve made use of this restriction on several occasions, namely so that we canassume:(1) All nontrivial holomorphic curves in R × M have at least one positive punc-ture;(2) The energy of a holomorphic curve in R × M can be bounded in terms ofits positive asymptotic orbits.It will be useful however for certain applications to permit a slightly wider class ofstable Hamiltonian structure. Recall that a hypersurface V in an almost complexmanifold ( W, J ) is called pseudoconvex if the maximal complex subbundle ξ := T V ∩ J ( T V ) ⊂ T V defines a contact structure on V whose canonical conformal symplectic bundle struc-ture tames J | ξ . For example, if α is a contact form on M and J ∈ J ( α ), then each ofthe hypersurfaces { const }× M is pseudoconvex in ( R × M, J ). The contact structure ξ induces an orientation on the hypersurface V ; if V comes with its own orientation(e.g. as a boundary component of W ), then we call it pseudoconvex if ξ is a positivecontact structure with respect to this orientation, and pseudoconcave otherwise.For example, if ( W, ω ) is a symplectic cobordism from ( M − , ξ − ) to ( M + , ξ + ) and J ∈ J ( W, ω, α + , α − ), then M + is pseudoconvex and M − is pseudoconcave. Definition . Given an odd-dimensional manifold M , we will say that analmost complex structure J on R × M is pseudoconvex if { r }× M is a pseudoconvexhypersurface in ( R × M, J ) for every r ∈ R , with the induced orientation such that ∂ r and { r } × M are positively transverse.If H = ( ω, λ ) is a stable Hamiltonian structure on M , then pseudoconvexity of J ∈ J ( H ) imposes conditions on H , in particular λ must be a contact form. It alsorequires J | ξ to be tamed by dλ | ξ , but unlike the case when J ∈ J ( λ ), J | ξ need not be compatible with it, i.e. the positive bilinear form dλ ( · , J · ) | ξ need not be symmetric.As always, J | ξ must be compatible with ω | ξ , but ω need not be an exact form for ectures on Symplectic Field Theory this to hold—the freedom to change [ ω ] ∈ H ( M ) will be the main benefit of thisgeneralization, particularly when we discuss weak symplectic fillings below. Proposition . Suppose H = ( ω, λ ) is a stable Hamiltonian structure ona closed manifold M and J ∈ J ( H ) is pseudoconvex. Then all nonconstant finite-energy J -holomorphic curves in R × M have at least one positive puncture, andtheir energies satisfy a uniform upper bound in terms of the periods of their positiveasymptotic orbits. Proof.
It is straightforward to check that either of the two proofs of Proposi-tion 10.9 given in Lecture 10 generalizes to any J on R × M that is pseudoconvex.In particular, pseudoconvexity implies that if u : ( ˙Σ , j ) → ( R × M, J ) is a J -holomorphic curve, then u ∗ dλ ≥
0, with equality only at points where u is tangentto ∂ r and the Reeb vector field. Stokes’ theorem thus gives(13.9) 0 ≤ Z ˙Σ u ∗ dλ = X z ∈ Γ + T z − X z ∈ Γ − T z , where T z > z ∈ Γ ± . Since J | ξ is also tamed by ω | ξ and ω annihilates the Reeb vectorfield, we similarly have u ∗ ω ≥
0, with the same condition for equality, and thecompactness of M then implies an estimate of the form0 ≤ u ∗ ω ≤ cu ∗ dλ for every J -holomorphic curve u : ( ˙Σ , j ) → ( R × M, J ), with a constant c > M , H and J . In light of (13.9), this implies an upper bound on R ˙Σ u ∗ ω in terms of the periods T z for z ∈ Γ + . Writing ω ϕ = ω + d ( ϕ ( r ) λ ) for suitable C -small increasing functions ϕ : R → R , we can then apply Stokes’ theorem to thesecond term in E ( u ) = sup ϕ Z ˙Σ u ∗ ω ϕ = Z ˙Σ u ∗ ω + sup ϕ Z ˙Σ u ∗ d ( ϕ ( r ) λ ) , implying a similar upper bound for E ( u ). (cid:3) Corollary . For any stable Hamiltonian structure H = ( ω, λ ) with anondegenerate Reeb vector field R H and a pseudoconvex J ∈ J ( H ) , one can useclosed R H -orbits and count J -holomorphic curves in R × M to define the chaincomplexes ( A [[ ~ ]] , D SFT ) , ( W , D H ) , ( P , d h ) and ( A , ∂ CH ) . We shall denote the homologies of the above chain complexes with coefficientsin R = Q [ H ( M ) /G ] by H SFT ∗ ( M, H , J ; R ) , H W ∗ ( M, H , J ; R ) , H RSFT ∗ ( M, H , J ; R ) , HC ∗ ( M, H , J ; R ) . We make no claim at this point about these homologies being invariant. For theexamples that we actually care about, this will turn out to be an irrelevant questiondue to Proposition 13.16 and Exercise 13.32 below.
Example . Suppose α is a contact form on ( M, ξ ) and H = (Ω , α ) is astable Hamiltonian structure. Then for all constants c > H c :=(Ω + c dα, α ) is also a stable Hamiltonian structure and there exists a pseudoconvex Chris Wendl J c ∈ J ( H c ). To see the latter, notice that H ′ c := (cid:0) c Ω + dα, α (cid:1) is another family ofstable Hamiltonian structures, with J ( H ′ c ) = J ( H c ) for all c , and H ′ c → ( dα, α ) as c → ∞ . Thus one can select J c ∈ J ( H c ) converging to some J ∞ ∈ J ( α ) as c → ∞ ,and these are pseudoconvex for c > J ∞ is. Proposition . In the setting of Example 13.15, assume α is nondegenerateand J ∞ ∈ J ( α ) is generic. If HC ∗ ( M, ξ ; R ) = 0 , then HC ∗ ( M, H c , J c ; R ) alsovanishes for all c > sufficiently large. Proof.
We will assume in the following that the usual (unrealistic) transver-sality assumptions hold, but the essential idea of the argument would not change inthe presence of abstract perturbations.Let ( A , ∂ ∞ CH ) denote the contact homology chain complex generated by closed R α -orbits, with ∂ ∞ CH counting J ∞ -holomorphic curves in R × M . The assumption HC ∗ ( M, ξ ; R ) = 0 means there exists an element f ∈ A with ∂ ∞ CH f = 1. Here f isa polynomial function of the q γ variables, and ∂ ∞ CH f counts a specific finite set ofFredholm regular index 1 curves in ( R × M, J ∞ ). Now let ( A , ∂ c CH ) denote the chaincomplex for HC ∗ ( M, H c , J c ; R ), and notice that since the stable Hamiltonian struc-tures ( dα, α ) and H c define matching Reeb vector fields, the set of generators is un-changed. There is also no change to this complex if we replace H c = (Ω + c dα, α ) by H ′ c = (cid:0) c Ω + dα, α (cid:1) : this changes the energies of individual J c -holomorphic curves,but the sets of finite-energy curves are still the same in both cases. We can assume J c → J ∞ in C ∞ as c → ∞ . The implicit function theorem then extends each ofthe finitely many J ∞ -holomorphic curves counted by ∂ ∞ f uniquely to a smooth 1-parameter family of J c -holomorphic curves for c > We claimthat these are the only curves counted by ∂ c CH f when c > c k → ∞ for which additional J c k -holomorphicindex 1 curves u k contribute to ∂ c k CH f , and since f has only finitely many terms rep-resenting possible positive asymptotic orbits, we can find a subsequence for which allthe u k have the same positive asymptotic orbits. A further subsequence then has allthe same negative asymptotic orbits as well since the Reeb flow is nondegenerate andthe total period of the negative orbits is bounded by the total period of the positiveorbits. Finally, since the sequence of stable Hamiltonian structures H ′ c k converges to( dα, α ), the curves u k have uniformly bounded energy with respect to H ′ c k , so thatSFT compactness yields a subsequence converging to a J ∞ -holomorphic building ofindex 1, which can only be one of the curves counted by ∂ ∞ CH f . This contradicts theuniqueness in the implicit function theorem and thus proves the claim. We concludethat for all c > ∂ c CH f = 1. (cid:3) Definition . Assume (
W, ω ) is a symplectic cobordism with stable bound-ary ∂W = − M − ⊔ M + , with induced stable Hamiltonian structures H ± = ( ω ± , λ ± )at M ± , and suppose J is an almost complex structure on the completion c W that is ω -tame on W and belongs to J ( H ± ) on the cylindrical ends. We will say that J In case you are concerned about the parametric moduli space being an orbifold instead of amanifold, just add asymptotic markers so that there is no isotropy, and divide by the appropriatecombinatorial factors to count. ectures on Symplectic Field Theory is pseudoconvex near infinity if the R -invariant almost complex structures J ± defined by restricting J to [0 , ∞ ) × M + and ( −∞ , × M − are both pseudoconvex.Note that the condition on J in the above definition can only be satisfied if λ ± are both positive contact forms on M ± , but the 2-forms ω ± need not be exact.Proving contact invariance of SFT requires counting curves in trivial exact sym-plectic cobordisms, but it is also natural to try to say things about non-exact strong symplectic cobordisms using SFT. These fit naturally into our previouslyestablished picture since every strong cobordism has collar neighborhoods near theboundary in which it matches the symplectization of a contact manifold. The fol-lowing more general notion of cobordism is also natural from a contact topologicalperspective, but fits less easily into the SFT picture.
Definition
MNW13 ]) . Given closed contact manifolds ( M + , ξ + ) and( M − , ξ − ) of dimension 2 n −
1, a weak symplectic cobordism from ( M − , ξ − ) to( M + , ξ + ) is a compact symplectic manifold ( W, ω ) with ∂W = − M − ⊔ M + admittingan ω -tame almost complex structure J for which the almost complex manifold ( W, J )is pseudoconvex at M + and pseudoconcave at M − , with ξ ± = T M ± ∩ J ( T M ± ) . Weak cobordisms are characterized by the existence of a tame almost complexstructure J whose restriction to ξ ± is tamed by two symplectic bundle structures, ω | ξ ± and dα ± | ξ ± (for any choices of contact forms α ± defining ξ ± ). Notice thatin dimension 4, the second condition is mostly vacuous, and the weak cobordismcondition just reduces to ω | ξ ± > . In this form, the low-dimensional case of Definition 13.18 has been around since thelate 1980’s, and there are many interesting results about it, e.g. examples of contact3-manifolds that are weakly but not strongly fillable [
Gir94 , Eli96 ]. We will see in § MNW13 ].One major difference between weak and strong cobordisms is that the latter arealways exact near the boundary, as the Liouville vector field is dual to a primitiveof ω . It turns out that up to deformation, weak fillings that are exact at theboundary are the same thing as strong fillings—this was first observed by Eliashbergin dimension three [ Eli91 , Prop. 3.1], and was extended to higher dimensions in[
MNW13 ]: Proposition . Suppose ( W, ω ) is a weak filling of a (2 n − -dimensionalcontact manifold ( M, ξ ) such that ω | T M is exact. Then after a homotopy of ω through If I were being hypercorrect about use of language, I might insist on saying that J is “pseu-doconvex near + ∞ and pseudoconcave near −∞ ,” as the orientation reversal at the negativeboundary makes M − technically a pseudoconcave hypersurface in ( c W , J ), not pseudoconvex. Butthis definition will only be useful to us in cases where M − = ∅ , so my linguistic guilt is limited. By strong cobordism , we mean the usual notion of a compact symplectic manifold with convexand/or concave boundary components (see § Chris Wendl a family of symplectic forms that vary only in a collar neighborhood of ∂W and defineweak fillings of ( M, ξ ) , ( W, ω ) is a strong filling of ( M, ξ ) . Proof.
Choose any contact form α for ξ , denote its Reeb vector field by R α ,and let Ω = ω | T M . Identify a collar neighborhood of ∂W in W smoothly with( − ǫ, × M , with the coordinate on ( − ǫ,
0] denoted by r , such that ∂ r and R α spanthe symplectic complement of ξ at ∂W and satisfy ω ( ∂ r , R α ) = 1. Then ω andΩ + d ( rα ) are cohomologous symplectic forms on ( − ǫ, × M that match at r = 0,hence a Moser deformation argument implies they are isotopic. We can thereforeassume without loss of generality that ω = Ω + d ( rα ) on the collar near ∂W .By assumption, Ω = dη for some 1-form η on M , and since ( W, ω ) is a weakfilling of (
M, ξ = ker α ), we can choose a complex structure J ξ on ξ that is tamedby both dα | ξ and dη | ξ . Now choose a smooth cutoff function β : [0 , ∞ ) → [0 ,
1] thathas compact support and equals 1 near 0. We claim that ω := d ( β ( r ) η ) + d ( rα )is a symplectic form on [0 , ∞ ) × M if | β ′ | is sufficiently small. Indeed, writing ω = dr ∧ ( α + β ′ ( r ) η ) + [ β ( r ) dη + r dα ], we have ω n = n dr ∧ α ∧ [ β ( r ) dη + r dα ] n − + nβ ′ ( r ) dr ∧ η ∧ [ β ( r ) dη + r dα ] n − . The first term is positive and bounded away from zero since dη | ξ and dα | ξ bothtame J ξ , hence do does β dη + r dα | ξ . The second term is then harmless if | β ′ | issufficiently small, proving ω n > c W = W ∪ M ([0 , ∞ ) × M ), and for each r ≥
0, the compact subdomains definedby r ≤ r define weak fillings of ( { r } × M, ξ ) since ω | ξ = ( β ( r ) dη + r dα ) | ξ alsotames J ξ . Notice that for r sufficiently large, the dη term disappears, so ω has aprimitive that restricts to { r } × M as a contact form for ξ , meaning we have a strong filling of this hypersurface. The desired deformation of ω can therefore bedefined by pulling back via a smooth family of diffeomorphisms ( − ǫ, → ( − ǫ, r ],where r varies from 0 to a sufficiently large constant. (cid:3) Unlike strong cobordisms, being a weak cobordism is an open condition: if (
W, ω )is a weak cobordism, then so is (
W, ω + ǫσ ) for any ǫ > σ , which need not be exact at ∂W . As a consequence, the cylindricalends of a completed weak cobordism cannot always be deformed to look like thesymplectization of a contact manifold. This is where Definition 13.17 comes inuseful. The proof of the next lemma is very much analogous to Proposition 13.19. Lemma
MNW13 , Lemma 2.10]) . Suppose ( W, ω ) is a weak filling ofa (2 n − -dimensional contact manifold ( M, ξ ) , α is a contact form for ξ and Ω is a closed -form on M with [Ω] = [ ω | T M ] ∈ H ( M ) . Then for any constant c > sufficiently large, after a homotopy of ω through a family of symplectic formsthat vary only in a collar neighborhood of ∂W and define weak fillings of ( M, ξ ) , ω | T M = Ω + c dα . (cid:3) The following result then provides a suitable model that can be used as Ω inthe above lemma when ω | T M is nonexact. The statement below is restricted to the ectures on Symplectic Field Theory case where [ ω | T M ] is a rational cohomology class; the reason for this is that it relieson a Donaldson-type existence result for contact submanifolds obtained as zero-setsof approximately holomorphic sections, due to Ibort, Marti´ınez-Torres and Presas[
IMTP00 ]. It seems likely that the rationality condition could be lifted with morework, and in dimension three this is known to be true; see [
NW11 , Prop. 2.6].
Lemma
CV15 , Prop. 2.18]) . For any rational cohomology class η ∈ H ( M ; Q ) on a closed (2 n − -dimensional contact manifold ( M, ξ ) , there existsa closed -form Ω and a nondegenerate contact form α for ξ such that (Ω , α ) is astable Hamiltonian structure. (cid:3) Combining all of the above results (including Example 13.15) proves:
Proposition . Suppose ( W, ω ) is a weak filling of a (2 n − -dimensionalcontact manifold ( M, ξ ) such that [ ω | T M ] ∈ H ( M ) is rational or n = 2 . Fix anondegenerate contact form α for ξ . Then there exists a closed -form Ω cohomol-ogous to ω | T M such that H := (Ω , α ) is a stable Hamiltonian structure, and forall c > sufficiently large, ω can be deformed in a collar neighborhood of ∂W ,through a family of symplectic forms defining weak fillings of ( M, ξ ) , to a new weakfilling for which ∂W is also stable and inherits the stable Hamiltonian structure H c := (Ω + c dα, α ) . In particular, after this deformation, the completed stable fill-ing admits a tame almost complex structure that is pseudoconvex near infinity andmay be assumed C ∞ -close to any given J ∈ J ( α ) . (cid:3) We will use this in § Remark . There is apparently no analogue of Propositions 13.19 and 13.22for negative boundary components of weak cobordisms, and this is one of a fewreasons why they are not often discussed. For example, if L is a Lagrangian torusin the standard symplectic 4-ball D , then the complement of a neighborhood of L in B defines a strong cobordism from the standard contact T to S . Thesymplectic form on this cobordism is obviously exact, but if any result analogous toProposition 13.19 were to hold at the concave boundary, then we could deform it toa Liouville cobordism. No such Liouville cobordism exists—it would imply that theLagrangian L ⊂ B is exact, thus violating Gromov’s famous theorem [ Gro85 ] onexact Lagrangians. curves. Fix a symplectic cobor-dism (
W, ω ) with stable boundary ∂W = − M − ⊔ M + carrying stable Hamiltonianstructures H ± = ( ω ± , λ ± ), along with a generic almost complex structure J that is ω -tame on W , belongs to J ( H ± ) on the cylindrical ends, and is pseudoconvex nearinfinity. This implies that the stabilizing 1-forms λ ± are both contact forms. Let usalso assume that the λ ± are both nondegenerate, and that the induced R -invariantalmost complex structures J ± ∈ J ( H ± ) are sufficiently generic to achieve regularityfor all holomorphic curves under consideration. In particular, these assumptionsmean that all the usual SFT chain complexes are well defined for ( M ± , H ± , J ± ; R ± )with any choice of coefficient ring R ± = Q [ H ( M ± ) /G ± ]. Denote the correspondingSFT generating functions by H ± . Chris Wendl
Recall from Lecture 12 that the auxiliary data on M + and M − includes a choiceof capping surface C γ for each closed Reeb orbit γ (or a capping chain with rationalcoefficients if H ( M ± ) has torsion). These surfaces satisfy ∂C γ = X i m i [ C ± i ] − [ γ ] , where the m i are integers and C ± i ⊂ M ± are fixed curves forming a basis of H ( M ± ).Assume H ( W ) is torsion free, in which case the same is true of H ( M + ) and H ( M − ). (Only minor modifications are needed if this assumption fails to hold,see Remark 13.1.) We can then fix the following additional auxiliary data:(1) A collection of reference curves S ∼ = C , . . . , C r ⊂ W whose homology classes from a basis of H ( W ).(2) A unitary trivialization of T W along each of the reference curves C , . . . , C r ,denoted collectively by τ .(3) A spanning surface S ± i for each of the positive/negative reference curves C ± i ⊂ M ± , i.e. a smooth map of a compact and oriented surface withboundary into W such that ∂S ± i = X j m ji [ C j ] − [ C ± i ]in the sense of singular 2-chains, where m ji ∈ Z are the unique coefficientswith [ C ± i ] = P j m ji [ C j ] ∈ H ( W ).Now to any collections of orbits γ ± = ( γ ± , . . . , γ ± k ± ) in M ± and a relative homol-ogy class A ∈ H ( W, ¯ γ + ∪ ¯ γ − ) with ∂A = P i [ γ + i ] − P j [ γ − j ], we can associate anabsolute homology class in two steps: first add A to suitable sums of the cappingsurfaces C γ ± i producing a 2-chain whose boundary is a linear combination of positiveand negative reference curves, then add a suitable linear combination of the S ± i sothat the boundary becomes the trivial linear combination of C , . . . , C r . With thisunderstood, we can now associate an absolute homology class[ u ] ∈ H ( W )to any asymptotically cylindrical J -holomorphic curve u : ( ˙Σ , j ) → ( c W , J ), andthis defines the notation M g,m ( J, A, γ + , γ − ) with A ∈ H ( W ). We now require thetrivializations of ξ ± along each C ± i to be compatible with τ in the sense that theyextend to trivializations of T W along the capping surfaces S ± i . With this convention,the Fredholm index formula takes the expected formind( u ) = ( n − χ ( ˙Σ) + 2 c ([ u ]) + k + X i =1 µ CZ ( γ i ) − k − X j =1 µ CZ ( γ j ) . If H ( W ) has torsion, then this whole discussion can be adapted as in § ectures on Symplectic Field Theory We will also need to impose a compatibility condition relating the coefficientrings R ± = Q [ H ( M ± ) /G ± ] to a corresponding choice on the cobordism W . Choosea subgroup G ⊂ H ( W ) such that(13.10) h [ ω ] , A i = 0 for all A ∈ G, and such that the maps H ( M ± ) → H ( W ) induced by the inclusions M ± ֒ → W send G ± into G . If [ ω ] = 0 ∈ H ( W ), then we will have to deal with noncompactsequences of J -holomorphic curves that have unbounded energy, so it becomes nec-essary to “complete” R to a Novikov ring R , which contains R but also includesinfinite formal sums ∞ X i =1 c i e A i such that h [ ω ] , A i i → + ∞ as i → ∞ . Note that the evaluation h [ ω ] , A i ∈ R is well defined for A ∈ H ( W ) /G due to(13.10).Analogously to our definition of H in Lecture 12, the generating function forindex 0 curves in c W is defined as a formal power series in the variables ~ , q γ (fororbits in M − ), and p γ (for orbits in M + ), with coefficients in R :(13.11) F = X u ∈M σ ( J ) ǫ ( u ) | Aut σ ( u ) | ~ g − e A q γ − p γ + , where M σ ( J ) denotes the moduli space of connected J -holomorphic curves u in c W with ind( u ) = 0 and only good asymptotic orbits, modulo permutations of thepunctures, and for each u : • g is the genus of u ; • A is the equivalence class of [ u ] ∈ H ( W ) in H ( W ) /G ; • γ ± = ( γ ± , . . . , γ ± k ± ) are the asymptotic orbits of u after arbitrarily fixingorderings of its positive and negative punctures; • ǫ ( u ) ∈ { , − } is the sign of u as a point in the 0-dimensional compo-nent of M $ ( J ) (after choosing an ordering of the punctures and asymptoticmarkers), relative to a choice of coherent orientations on M $ ( J ).As usual, the product ǫ ( u ) q γ − p γ + is independent of choices. We shall regard F asan element in an enlarged operator algebra that includes q and p variables for goodorbits in both M + and M − , related to each other by the supercommutation relations[ p γ − , q γ + ] = [ p γ + , q γ − ] = [ q γ − , q γ + ] = [ p γ − , p γ + ] = 0whenever γ − is an orbit in M − and γ + is an orbit in M + . Since all curves countedby F have index 0, F is homogeneous with degree | F | = 0 . Notice that for any fixed monomial q γ − p γ + , the corresponding set of curves in M σ ( J )may be infinite if ω is nonexact, but SFT compactness implies that the set of suchcurves with any given bound on R ˙Σ u ∗ ω is bounded. As a consequence, the coefficientof q γ − p γ + in F belongs to the Novikov ring R . Chris Wendl
Consider next the series exp( F ) := ∞ X k =0 k ! F k . We will be able to view this as a formal power series in q and p variables and a formalLaurent series in ~ with coefficients in R , though it is not obvious at first glancewhether its coefficients are in any sense finite. We will deduce this after interpretingit as a count of disconnected index 0 curves: first, writeexp( F ) = ∞ X k =0 k ! X ( u ,...,u k ) ∈ ( M σ ( J )) k ǫ ( u ) . . . ǫ ( u k ) | Aut σ ( u ) | . . . | Aut σ ( u k ) | ~ g + ... + g k − k e A + ...A k · q γ − p γ +1 . . . q γ − k p γ + k ! . Observe that since each of the curves u i ∈ M σ ( J ) in this expansion has index 0,the monomials q γ − i p γ + i all have even degree and thus the order in which they arewritten does not matter. Now for a given collection of distinct curves v , . . . , v N andintegers k , . . . , k N ∈ N with k + . . . + k N = k , the various permutations of( u , . . . , u k ) := ( v , . . . , v | {z } k , . . . , v N , . . . , v N | {z } k N ) ∈ ( M σ ( J )) k occur k ! k ! ...k N ! times in the above sum, so if we forget the ordering, then the contri-bution of this particular k -tuple of curves to exp( F ) is ǫ ( u ) . . . ǫ ( u k ) k ! . . . k N ! | Aut σ ( u ) | . . . | Aut σ ( u k ) | ~ g + ... + g k − k e A + ... + A k q γ − p γ +1 . . . q γ − k p γ + k . Notice next that the denominator k ! . . . k N ! | Aut σ ( u ) | . . . | Aut σ ( u k ) | is the order ofthe automorphism group of the disconnected curve formed by the disjoint union of u , . . . , u k : the extra factors k i ! come from automorphisms that permute connectedcomponents of the domain. Thus exp( F ) can also be written as in (13.11), but with M σ ( J ) replaced by the moduli space of potentially disconnected index 0 curves withunordered punctures, and g − g + . . . + g k − k for any curve that has k connected components of genera g , . . . , g k . One subtlety that was glossed over inthe above discussion: the sum also includes the unique curve with zero components,i.e. the “empty” J -holomorphic curve, which appears as the initial 1 in the seriesexpansion of exp( F ).With this interpretation of exp( F ) understood, we can now address the possibil-ity that the infinite sum defining exp( F ) might include infinitely many terms for agiven monomial ~ m q γ − p γ + , i.e. that there are infinitely many disconnected index 0curves with fixed asymptotic orbits and a fixed sum of the genera minus the numberof connected components. We claim that this can indeed, happen, but only if thecurves belong to a sequence of homology classes A i ∈ H ( M ) /G with h [ ω ] , A i i → ∞ ,hence the coefficient of ~ m q γ − p γ + in exp( F ) belongs to the Novikov ring R . Thedanger here comes only from closed curves, since a disjoint union of two curveswith punctures always has strictly more punctures. Notice also that for any given ectures on Symplectic Field Theory tuples of orbits γ ± , there exists a number c ∈ R depending only on these orbits andthe chosen capping surfaces such that every (possibly disconnected) J -holomorphiccurve u : ˙Σ → c W asymptotic to γ ± satisfies h [ ω ] , [ u ] i ≥ c. This follows from the fact that the integral of ω over the relative homology class of u always has a nonnegative integrand. Lemma . Given constants C ∈ R and k ∈ Z , there exists a number N ∈ N such that if u : (Σ , j ) → ( c W , J ) is a closed J -holomorphic curve satisfying R Σ u ∗ ω ≤ C , with m connected components of genera g , . . . , g m satisfying g + . . . + g m − m = k ,then m ≤ N . Proof.
Note first that for each integer g ≥
0, there is an energy thresh-old , i.e. a constant c g > J -holomorphic curve u : Σ → c W of genus g has Z Σ u ∗ ω ≥ c g . This is an easy consequence of SFT compactness: indeed, if there were no suchconstant, then we would find a sequence u k : Σ → c W of connected closed curveswith genus g such that E ( u k ) = Z Σ u ∗ ω → R Σ u ∗ ω ϕ depends only on the homologyclass of u in order to simplify the usual definition of energy for asymptoticallycylindrical curves. SFT compactness then gives a subsequence of u k that convergesto a stable holomorphic building in which every component has zero energy and istherefore constant. Since there are no marked points in the picture, no such buildingexists, so this is a contradiction.Now if u is a disconnected curve satisfying the stated conditions, the bound on R Σ u ∗ ω combines with the energy threshold to give a bound for each g ≥ u with genus g . In particular, there is a boundon the number of components with genus 0 or 1. All other components contributepositively to the left hand side of the relation P mi =1 ( g i −
1) = k , so this implies auniversal bound on m . (cid:3) Corollary . Fix constants C ∈ R and k ∈ Z , and tuples of Reeb orbits γ ± , and assume that the usual transversality conditions hold. Then there exist atmost finitely many potentially disconnected J -holomorphic curves u : ˙Σ → c W withindex such that the number of connected components m and the genera g , . . . , g m of its components satisfy g + . . . + g m − m = k . Corollary . The expression exp( F ) is a formal power series in q and p variables and a formal Laurent series in ~ , with coefficients in the Novikov ring R . Chris Wendl
The necessity of considering disconnected curves becomes clear when one tries totranslate the compactness and gluing theory of J -holomorphic curves in c W into alge-braic relations. In particular, consider the 1-dimensional moduli space of connectedindex 1 curves in c W with genus g . The boundary points of the compactification ofthis space consist of two types of buildings: Type 1 : A main level of index 0 and an upper level of index 1;
Type 2 : A main level of index 0 and a lower level of index 1.This is clear under the usual transversality assumptions since regular curves in c W must have index at least 0, while regular curves in the symplectizations R × M ± have index at least 1 unless they are trivial cylinders. The building must also beconnected and have arithmetic genus g , but there is nothing to guarantee that eachindividual level is connected. In fact, we already saw this issue in Lecture 12 whenproving H = 0, but it was simpler to deal with there, because disconnected regularcurves of index 1 in a symplectization always have a unique nontrivial component,while the rest are trivial cylinders. In the cobordism c W , on the other hand, adisconnected index 0 curve can be formed by any disjoint union of index 0 curves,all of which are nontrivial. Exponentiation provides a convenient way to encode alldata about disconnected curves in terms of connected curves.Since the union of all buildings of types 1 and 2 described above forms theboundary of a compact oriented 1-manifold, the count of these buildings is zero,and this fact is encoded in the so-called master equation (13.12) H − exp( F ) | p − =0 − exp( F ) H + | q + =0 = 0 , where the expressions “ p − = 0” and “ q + = 0” mean that we discard all terms in H − exp( F ) − exp( F ) H + containing any variables p γ for orbits in M − or q γ for orbitsin M + . The resulting expression is therefore a formal power series in q variablesfor orbits in M − and p variables for orbits in M + , representing a count of generallydisconnected index 1 holomorphic buildings in c W with the specified asymptotics.The various ways to form such buildings by choices of gluings is again encoded bythe commutator algebra. The master equation (13.12) can be used to prove thechain map property for counts of curves in cobordisms, thus it is an essential pieceof the invariance proof for each of the homology theories introduced above. Exercise . Fill in the details of the proof of (13.12). BV ∞ -algebra In this section we discuss the specific theory H SFT ∗ ( M, ξ ; R ), defined as the ho-mology of the chain complex ( A [[ ~ ]] , D SFT ). The case G = H ( M ) with trivial groupring coefficients Q [ H ( M ) /G ] = Q will be abbreviated as H SFT ∗ ( M, ξ ) := H SFT ∗ ( M, ξ ; Q ) . As we defined it, D SFT acts on A [[ ~ ]] by treating the generating function H as adifferential operator via the substitution(13.13) p γ = κ γ ~ ∂∂q γ . ectures on Symplectic Field Theory According to [
CL09 ], this makes ( A [[ ~ ]] , D SFT ) into a BV ∞ -algebra; we’ll have noparticular need to discuss here what that means, but one convenient feature is theexpansion(13.14) D SFT = 1 ~ ∞ X k =1 D ( k )SFT ~ k , in which each D ( k )SFT : A → A is a differential operator of order ≤ k (see [ CL09 , § k ∈ N , D ( k )SFT is a count of all index 1 holomorphic curves that have genus g ≥ m ≥ g + m = k . In particular, D (1)SFT is simply the contact homology differential ∂ CH , and the expansion (13.14) impliestogether with D = 0 that ( D (1)SFT ) = 0, hence we again see the chain complexfor contact homology hidden inside a version of the “full” SFT complex. One can use the master equation(13.12) to prove invariance of H SFT ∗ ( M, ξ ; R ) by a straightforward generalization ofthe usual Floer-theoretic argument. Suppose ( W, dλ ) is an exact symplectic cobor-dism from ( M − , ξ − ) to ( M + , ξ + ) with λ | T M ± = α ± , and choose a generic almostcomplex structure J on c W that is dλ -compatible on W and restricts to the cylin-drical ends as generic elements J ± ∈ J ( α ± ). Let ( A ± [[ ~ ]] , D ± SFT ) denote the chaincomplexes associated to the data ( α ± , J ± ), and for simplicity in this initial discus-sion, choose the trivial coefficient ring R = Q for both. We then define a map Φ : A + [[ ~ ]] → A − [[ ~ ]] : f exp( F ) f | q + =0 , where the generating function exp( F ) is regarded as a differential operator via thesubstitution (13.13), with e A := 1 for all A ∈ H ( W ) since we are using trivialcoefficients, and “ q + = 0” means that after applying exp( F ) to change f into afunction of q variables for orbits in both M + and M − , we discard all terms thatinvolve orbits in M + . The exactness of the cobordism implies that negative powersof ~ do not appear in Φf , thus producing an element of A − [[ ~ ]]: indeed, since thereare no holomorphic curves in c W without positive punctures, every term in F containsat least one p variable, so that negative powers of ~ do not appear in exp( F ) afterapplying (13.13).The master equation for F now translates into the fact that Φ is a chain map, D − SFT ◦ Φ − Φ ◦ D +SFT , thus it descends to homology. The geometric meaning of Φ is straightforward todescribe: analogous to (12.11) in Lecture 12, we can write(13.15) Φ q γ = ∞ X g =0 X γ ′ ~ g + k − n g ( γ , γ ′ , k ) q γ ′ , where n g ( γ , γ ′ , k ) is a product of some combinatorial factors with a signed countof disconnected index 0 holomorphic curves with connected components of genus g , . . . , g m satisfying g + . . . + g m − m = g −
1, and with positive ends at γ andnegative ends at γ ′ , where k is the number of positive ends. Chris Wendl
Let’s discuss two applications of the cobordism map Φ . First, note that if W is a trivial symplectic cobordism [0 , × M , then the above discussion can easily be gen-eralized with ( A ± , D ± SFT ) both defined over the same group ring R = Q [ H ( M ) /G ]for any choice of G ⊂ H ( M ). There is no need to consider a Novikov ring indefining F here since the cobordism is exact. We therefore obtain a chain mapwith arbitrary group ring coefficients, and extending this discussion along standardFloer-theoretic principles will imply that the chain map is an isomorphism: this canbe used in particular to prove that H SFT ∗ ( M, ξ ; R ) does not depend on the choices ofcontact form and almost complex structure. There are two additional steps involvedin this argument: first, one needs to use a chain homotopy to prove that Φ doesnot depend on the choice of almost complex structure J on c W . Given a generichomotopy { J s } s ∈ [0 , , the chain homotopy map Ψ : A + [[ ~ ]] → A − [[ ~ ]]is defined as a differential operator in the same manner as Φ , but counting pairs( s, u ) where s ∈ [0 ,
1] is a parameter value for which J s is nongeneric and u is adisconnected J s -holomorphic curve in c W with index −
1. We saw how this worksfor cylindrical contact homology in Lecture 10, but there is a new subtlety nowthat should be mentioned: in principle, a disconnected index − c W couldhave arbitrarily many components, including perhaps many with index − s, u ) for J s -holomorphic curves u of index 0 may include buildings thathave symplectization levels of index greater than 1, balanced by disjoint unions ofmany index − s, u ) where u is a connected J s -holomorphic curve with index −
1, one can (iftransversality is achievable at all) use a genericity argument to assume without lossof generality that for any given s ∈ [0 , one connected index − − q γ − p γ + that contains an odd number ofodd generators, and any nontrivial product of such generators therefore disappears in A since odd generators anticommute with themselves. This algebraic miracleencodes a convenient fact about coherent orientations: whenever one of the horriblebuildings described above appears, one can reorder two of the index − R -invariant data gives a cobordismmap that just counts trivial cylinders and is therefore the identity, it follows that ectures on Symplectic Field Theory cobordism maps relating different pairs of data ( α ± , J ± ) are always invertible, andthis proves the invariance of H SFT ∗ ( M, ξ ; R ).The second application concerns nontrivial exact cobordisms, and it is immediatefrom the fact that Φ is a chain map: Theorem . Any exact cobordism ( W, dλ ) from ( M − , ξ − ) to ( M + , ξ + ) givesrise to a Q [[ ~ ]] -linear map H SFT ∗ ( M + , ξ + ) → H SFT ∗ ( M − , ξ − ) . (cid:3) It is much more complicated to say what happens in the event of a nonexactcobordism, but slightly easier if we restrict our attention to fillings, i.e. the case with M − = ∅ . Assume ( W, ω ) is a compact symplectic manifold with stable boundary M ,inheriting a stable Hamiltonian structure H = (Ω , α ) for which α is a nondegeneratecontact form, and assume also that the completion c W admits an almost complexstructure J that is ω -tame on W and has a pseudoconvex restriction J + ∈ J ( H ) tothe cylindrical end. We saw in Proposition 13.22 that these conditions can alwaysbe achieved for a weak filling after deforming the symplectic structure. Let G := ker[ ω ] := { A ∈ H ( W ) | h [ ω ] , A i = 0 } , and choose G + ⊂ H ( M ) to be any subgroup such that the map H ( M ) → H ( W )induced by the inclusion M ֒ → W sends G + into G . In other words, G + can be anysubgroup of ker[Ω] ⊂ H ( M ). Define the group rings R + = Q [ H ( M ) /G + ] , R = Q [ H ( W ) / ker[ ω ]] , with the Novikov completion of R denoted by R . The map H ( M ) /G + → H ( W ) /G induced by M ֒ → W then gives a natural ring homomorphism(13.16) R + → R. If ω is not exact, then it may no longer be true that every term in F has at leastone p variable. Let us write F = F + F , where F contains no p variables and F = O ( p ), i.e. F counts all closed curvesin c W , and F counts everything else. Since F and F have even degree, theycommute, and thus exp( F ) = exp( F ) exp( F ) . where exp( F ) is an invertible element of R [[ ~ , ~ − ]] since exp( − F ) exp( F ) = 1.By the master equation, exp( F ) exp( F ) H = O ( q ) , hence exp( F ) H = exp( − F ) O ( q ) = O ( q ) since exp( − F ) contains no p variables.Using the substitution (13.13), and using (13.16) to map coefficients in R + to R , itfollows that exp( F ) gives rise to a differential operator Φ : A [[ ~ ]] → R [[ ~ ]] : f exp( F ) f | q =0 , Chris Wendl which is a chain map to the SFT of the empty set with Novikov coefficients, meaning Φ ◦ D SFT = 0 . This chain map counts the disconnected index 0 curves in c W whose connectedcomponents all have at least one positive puncture. Theorem . Suppose ( W, ω ) is a compact symplectic manifold with sta-ble boundary ( M, H = (Ω , α )) , where α is a nondegenerate contact form, and itscompletion c W admits an almost complex structure that is ω -tame on W and hasa generic and pseudoconvex restriction J + ∈ J ( H ) to the cylindrical end. Let R denote the Novikov completion of Q [ H ( W ) / ker[ ω ]] , and let R + = Q [ H ( M ) /G + ] ,where G + ⊂ H ( M ) is any subgroup on which the evaluation of [Ω] ∈ H ( M ) vanishes. Then there exists an R [[ ~ ]] -linear map H SFT ∗ ( M, H , J + ; R + ) → R [[ ~ ]] . (cid:3) We can now generalize the notion of algebraicovertwistedness. Notice that since every term in D SFT is a differential operatorof order at least 1, D SFT f = 0 for all f ∈ R [[ ~ ]] , hence every element of the extended coefficient ring R [[ ~ ]] represents an elementof H SFT ∗ ( M, ξ ; R ) that may or may not be trivial. Since D SFT commutes with allelements of R [[ ~ ]], the subset consisting of elements that are trivial in homologyforms an ideal. The following definition originates in [ LW11 ]. Definition . We say that a closed contact manifold (
M, ξ ) has algebraictorsion of order k (or k -torsion for short) with coefficients in R if[ ~ k ] = 0 ∈ H SFT ∗ ( M, ξ ; R ) . The numerical invariant AT(
M, ξ ; R ) ∈ N ∪ { , ∞} is defined to be the smallest integer k such that ( M, ξ ) has algebraic k -torsion butno ( k − ∞ if there is no algebraic torsion of any order.Several consequences of algebraic torsion can be read off quickly from the prop-erties of SFT cobordism maps. Consider first the case of trivial coefficients R = Q ,which we shall refer to as untwisted algebraic torsion and abbreviateAT( M, ξ ) := AT(
M, ξ ; Q ) . If (
W, ω ) is a strong filling of (
M, ξ ), then the hypotheses of Theorem 13.29 arefulfilled even with G + = H ( M ) since ω is exact at the boundary, thus we obtaina Q [[ ~ ]]-linear map H SFT ∗ ( M, ξ ) → R [[ ~ ]], with R denoting the Novikov completionof Q [ H ( W ) / ker[ ω ]]. If [ ~ k ] = 0 ∈ H SFT ∗ ( M, ξ ), then the cobordism map implies acontradiction since ~ k does not equal 0 in R [[ ~ ]]. Similarly, if ( W, dλ ) is an exactcobordism from ( M − , ξ − ) to ( M + , ξ + ), then the cobordism map H SFT ∗ ( M + , ξ + ) → H SFT ∗ ( M − , ξ − ) of Theorem 13.28 is also Q [[ ~ ]]-linear, and thus any algebraic k -torsionin ( M + , ξ + ) is inherited by ( M − , ξ − ). This proves: ectures on Symplectic Field Theory Theorem . Contact manifolds with
AT(
M, ξ ) < ∞ are not strongly fillable.Moreover, if there exists an exact symplectic cobordism from ( M − , ξ − ) to ( M + , ξ + ) ,then AT( M − , ξ − ) ≤ AT( M + , ξ + ) . (cid:3) It is known (see [
Wen13 ]) that the second part of the above theorem does nothold for strong symplectic cobordisms in general, so exactness of cobordisms is ameaningful symplectic topological condition, not just a technical hypothesis. It isalso known thanks to a construction of Ghiggini [
Ghi05 ] that strong and exactfillability are not equivalent conditions, but Ghiggini’s proof of this uses HeegaardFloer homology; thus far it is not known whether this phenomenon can be detectedvia SFT or other holomorphic curve techniques.There are also many known examples of contact manifolds that have untwistedalgebraic torsion but are weakly fillable. The simplest are the tight tori ( T , ξ k )for k ≥
2, for which weak fillings were first constructed by Giroux [
Gir94 ], butEliashberg [
Eli96 ] showed that strong fillings do not exist, and we will see in Lec-ture 16 that AT( T , ξ k ) = 1. The weak/strong distinction can often be detected viathe choice of coefficients in SFT. We saw in § M, ξ ) can always be deformed so as to have stable boundary with data( H = (Ω , α ) , J + ) for which α is a nondegenerate contact form and J + is C ∞ -close toany given element of J ( α ). Proposition 13.16 showed that if ( M, ξ ) is algebraicallyovertwisted, then the contact homology for the stable Hamiltonian data ( H , J + ) canalso be made to vanish. Exercise . Generalize the proof of Prop. 13.16 to show that if (
M, ξ ) hasalgebraic k -torsion with coefficients in R , then also [ ~ k ] = 0 ∈ H SFT ∗ ( M, H c , J c ; R )for sufficiently large c > M, ξ ) has fullytwisted algebraic k -torsion whenever [ ~ k ] = 0 ∈ H SFT ∗ ( M, ξ ; Q [ H ( M )]). Note thatin parallel with Remark 13.10, any nested pair of subgroups G ⊂ G ′ ⊂ H ( M ) givesrise to a map H SFT ∗ ( M, ξ ; Q [ H ( M ) /G ′ ]) → H SFT ∗ ( M, ξ ; Q [ H ( M ) /G ]) , which is a morphism in the sense that it maps the unit and all powers of ~ tothemselves. This implies that ( M, ξ ) has fully twisted k -torsion if and only if it has k -torsion for every choice of coefficients. Theorem . If ( M, ξ ) is a closed contact manifold with a finite order ofalgebraic torsion with coefficients in R = Q [ H ( M ) /G ] for some subgroup G , then ( M, ξ ) does not admit any weak symplectic filling ( W, ω ) for which [ ω | T M ] ∈ H ( M ) is rational and annihilates all elements of G . In particular, if ( M, ξ ) has fully twistedalgebraic torsion of some finite order, then it is not weakly fillable. Remark . The rationality condition in Theorem 13.33 can probably belifted, and is known to be unnecessary at least in dimension three. It is clear in anycase that if (
M, ξ ) admits a weak filling (
W, ω ), then one can always make a smallperturbation of ω to produce a weak filling for which [ ω | T M ] ∈ H ( M ; Q ). Chris Wendl
We will see some concrete examples of algebraic torsion computations in Lec-ture 16. Let us conclude this discussion for now with the observation that algebraictorsion of order zero is a notion we’ve seen before:
Proposition . For any closed contact manifold ( M, ξ ) and group ring R = Q [ H ( M ) /G ] , the following conditions are equivalent:(1) ( M, ξ ) has algebraic -torsion (with coefficients in R );(2) ( M, ξ ) is algebraically overtwisted (with coefficients in R );(3) H SFT ∗ ( M, ξ ; R ) = 0 . Proof.
It is obvious that (3) implies (1). Since D SFT f = ∂ CH f + O ( ~ ) for f ∈ A ,the R [[ ~ ]]-linear map A [[ ~ ]] → A : F F | ~ =0 defines a chain map ( A [[ ~ ]] , D ) → ( A , ∂ CH ) and thus descends to an R [[ ~ ]]-linearmap H SFT ∗ ( M, ξ ; R ) → HC ∗ ( M, ξ ; R ). The existence of this map proves that (1)implies (2).To prove that (2) implies (3), recall first that if there exists f ∈ A with ∂ CH f =1, then the fact that HC ∗ ( M, ξ ; R ) = 0 follows easily since for any g ∈ A with ∂ CH g = 0, the graded Leibniz rule implies ∂ CH ( fg ) = ( ∂ CH f ) g − f ( ∂ CH g ) = g . Thisworks because ∂ CH is a derivation—but D SFT is not one, so the same trick will notquite work for D SFT . The trick in proving H SFT ∗ ( M, ξ ; R ) = 0 will be to quantifythe failure of D SFT to be a derivation. For our purposes, it suffices to know that(13.17) D SFT ( FG ) = ( D SFT F ) G + ( − | F | F ( D SFT G ) + O ( ~ )holds for all F , G ∈ A [[ ~ ]], which follows from the fact that ∂ CH is a derivation.With this remark out of the way, suppose f ∈ A satisfies ∂ CH f = 1, in which case(13.18) D SFT f = 1 + ~ G for some G ∈ A [[ ~ ]]. We claim then that for any Q ∈ A [[ ~ ]] with D SFT Q = 0, thereexists Q ∈ A [[ ~ ]] with(13.19) D SFT ( fQ ) = Q + ~ Q and D SFT Q = 0. Indeed, (13.19) follows from (13.17) and (13.18) since D SFT Q = 0,and D SFT Q = 0 then follows by applying D SFT to (13.19) and using D = 0.Fixing Q := Q ∈ A [[ ~ ]], we can now define a sequence Q k ∈ A [[ ~ ]] satisfying D SFT Q k = 0 for all integers k ≥ D SFT ( fQ k ) = Q k + ~ Q k +1 . Then P ∞ k =0 ( − k ~ k Q k ∈ A [[ ~ ]], and D SFT f ∞ X k =0 ( − k ~ k Q k ! = Q . (cid:3) ECTURE 14
Transversality and embedding controls in dimension four
The final three lectures will be included in the published version of this book.For updates on publication, see the author’s website
ECTURE 15
Intersection theory for punctured holomorphic curves
The final three lectures will be included in the published version of this book.For updates on publication, see the author’s website
ECTURE 16
Torsion computations and applications
The final three lectures will be included in the published version of this book.For updates on publication, see the author’s website
PPENDIX A
Sobolev spacesContents
A.1. Approximation, extension and embedding theorems 301A.2. Products, compositions, and rescaling 305A.3. Spaces of sections of vector bundles 311A.4. Some remarks on domains with cylindrical ends 316
In this appendix, we review some of the standard properties of Sobolev spaces,in particular using them to prove Propositions 2.7, 2.8 and 2.10 from § AF03 ]. A.1. Approximation, extension and embedding theorems
Unless otherwise noted, all functions in the following are assumed to be definedon a nonempty open subset
U ⊂ R n with its standard Lebesgue measure, and taking values in a finite-dimensional normedvector space that will usually not need to be specified, though occasionally we willassume it is R or C so that one can define products of functions. The domain U will also sometimes have additional conditions specified such as boundedness or reg-ularity at the boundary, though we will try not to add too many more restrictionsthan are really needed. The most useful assumption to impose on U is known asthe strong local Lipschitz condition : if U is bounded, then it means simply thatnear every boundary point of U , one can find smooth local coordinates in which U looks like the region bounded by the graph of a Lipschitz-continuous function,and in this case we call U a bounded Lipschitz domain . If U is unbounded,then one needs to impose extra conditions guaranteeing e.g. uniformity of Lipschitzconstants, and the precise definition becomes a bit lengthy (see [ AF03 , § , × (0 , ∞ ) ⊂ R which have smooth boundarywith finitely many corners. We will repeatedly need to use the generalized versionof H¨older’s inequality , which states that for any finite collection of measurable
Chris Wendl functions f , . . . , f m ,(A.1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m Y i =1 | f i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ m Y i =1 k f i k L pi for 1 ≤ p ≤ p , . . . , p m ≤ ∞ with 1 p = m X i =1 p i . This is an easy corollary of the standard version, (cid:13)(cid:13) | f | · | g | (cid:13)(cid:13) L ≤ k f k L p · k g k L q whenever 1 ≤ p, q ≤ ∞ and 1 = 1 p + 1 q . For an integer k ≥ p ∈ [1 , ∞ ] we define W k,p ( U ) as in § f ∈ L p ( U ) which have weak partial derivatives ∂ α f ∈ L p ( U )for all | α | ≤ k . For p = 2, these spaces are also often denoted by H k ( U ) := W k, ( U ) , and they admit Hilbert space structures with inner product h f, g i H k = X | β |≤ k h ∂ α f, ∂ α g i L . We denote by W k,p ( U ) ⊂ W k,p ( U ) , H k ( U ) ⊂ H k ( U )the closed subspaces defined as the closures of C ∞ ( U ) with respect to the relevantnorms. Since C ∞ ( U ) is dense in L p ( U ) for 1 ≤ p < ∞ (see e.g. [ LL01 , § W ,p ( U ) and W ,p ( U ) for p < ∞ , but in general W k,p ( U ) = W k,p ( U ) for k ≥
1, with a few notable exceptions such as the case U = R n (cf. Corollary A.2 below). Let W k,p loc ( U ) := (cid:8) functions f on U (cid:12)(cid:12) f ∈ W k,p ( V ) for all open subsets V ⊂ U with compact closure
V ⊂ U (cid:9) , and we say that a sequence f j ∈ W k,p loc ( U ) converges in W k,p loc to f ∈ W k,p loc ( U ) ifthe restrictions to all precompact open subsets V ⊂ V ⊂ U converge in W k,p ( V ).Recall that for k ∈ { , , , . . . , ∞} , C k ( U ) denotes the space of functions on U withcontinuous derivatives up to order k , while C k ( U ) ⊂ C k ( U )is the space of f ∈ C k ( U ) such that for all | α | ≤ k , ∂ α f is bounded and uniformlycontinuous. Theorem
A.1 ([
AF03 , § . For any open subset
U ⊂ R n , and any k ≥ , ≤ p < ∞ , the subspace C ∞ ( U ) ∩ W k,p ( U ) ⊂ W k,p ( U ) is dense. Moreover, if U ⊂ R n satisfies the strong local Lipschitz condition, then thespace n f ∈ C ∞ ( U ) (cid:12)(cid:12)(cid:12) f = ˜ f | U for some ˜ f ∈ C ∞ ( R n ) o is also dense in W k,p ( U ) , so in particular, C ∞ ( U ) ∩ W k,p ( U ) ⊂ W k,p ( U ) ectures on Symplectic Field Theory is dense. (cid:3) Corollary
A.2 . The space C ∞ ( R n ) is dense in W k,p ( R n ) for every k ≥ and p ∈ [1 , ∞ ) . (cid:3) Here is another useful characterization of W k,p ( U ): Theorem
A.3 ([
AF03 , § . Assume
U ⊂ R n is an open subset satisfying thestrong local Lipschitz condition. Then a function f ∈ W k,p ( U ) belongs to W k,p ( U ) if and only if the function ˜ f on R n defined to match f on U and everywhere elsebelongs to W k,p ( R n ) . (cid:3) While it is obvious from the definitions that functions in W k,p ( U ) always admitextensions of class W k,p over R n , this is much less obvious for functions in W k,p ( U )in general, and it is not true without sufficient assumptions about the regularityof ∂ U . For our purposes it suffices to consider the following case. Theorem
A.4 ([
AF03 , § . Assume
U ⊂ R n is a bounded open subset suchthat ∂ U is a submanifold of class C m for some m ∈ { , , , . . . , ∞} . Then thereexists a linear operator E that maps functions defined almost everywhere on U tofunctions defined almost everywhere on R n and has the following properties: • For every function f on U , Ef | U ≡ f almost everywhere; • For every nonnegative integer k ≤ m and every p ∈ [1 , ∞ ) , E defines abounded linear operator W k,p ( U ) → W k,p ( R n ) . (cid:3) Corollary
A.5 . Suppose U , U ′ ⊂ R n are open subsets such that U has compactclosure contained in U ′ . If U satisfies the hypothesis of Theorem A.4, then theresulting extension operator E can be chosen such that it maps each W k,p ( U ) for k ≤ m and ≤ p < ∞ into W k,p ( U ′ ) . Proof.
Choose a smooth function ρ : U ′ → [0 ,
1] that has compact support andequals 1 on U , then replace the operator E given by Theorem A.4 with the operator f ρ · Ef . (cid:3) To state the Sobolev embedding theorem in its proper generality, recall that for0 < α ≤
1, the
H¨older seminorm of a function f on U is defined by | f | C α := | f | C α ( U ) := sup x = y ∈U | f ( x ) − f ( y ) || x − y | α , and C k,α ( U ) is then defined as the Banach space of functions f ∈ C k ( U ) for whichthe norm k f k C k,α := k f k C k + max | β | = k | ∂ β f | C α is finite. In reading the following statement, it is important to remember thatelements of W k,p ( U ) are technically not functions, but rather equivalence classes of functions defined almost everywhere. Thus when we say e.g. that there is aninclusion W k,p ( U ) ֒ → C m,α ( U ), the literal meaning is that for every function f representing an element of W k,p ( U ), one can change the values of f in a unique way Chris Wendl on some set of measure zero in U so that after this change, f ∈ C m,α ( U ). Continuityof the inclusion means that there is a bound of the form k f k C m,α ≤ c k f k W k,p for all f ∈ W k,p ( U ), where c > m , α , k , p and U , but not on f . Theorem
A.6 ([
AF03 , § . Assume
U ⊂ R n is an open subset satisfyingthe strong local Lipschitz condition, k ≥ is an integer and ≤ p < ∞ .(1) If kp > n and k − n/p < , then there exist continuous inclusions W k,p ( U ) ֒ → C ,α ( U ) for each α ∈ (0 , k − n/p ] ,W k,p ( U ) ֒ → L q ( U ) for each q ∈ [ p, ∞ ] . (2) If kp < n and p ∗ > p is defined by the condition p ∗ = 1 p − kn , then there exist continuous inclusions W k,p ( U ) ֒ → L q ( U ) , for each q ∈ [ p, p ∗ ] . (3) If kp = n , then there exist continuous inclusions W k,p ( U ) ֒ → L q ( U ) , for each q ∈ [ p, ∞ ) . Moreover, the spaces W k,p ( U ) admit similar inclusions under no assumption on theopen subset U ⊂ R n . (cid:3) Under the same assumption on the domain U , one can apply Theorem A.6 to suc-cessive derivatives of functions in W k,p ( U ) and thus obtain the following inclusionsfor any integer d ≥ W k + d,p ( U ) ֒ → C d,α ( U ) if kp > n and 0 < α ≤ k − n/p < , (A.3) W k + d,p ( U ) ֒ → W d,q ( U ) if kp > n and p ≤ q ≤ ∞ , (A.4) W k + d,p ( U ) ֒ → W d,q ( U ) if kp < n and p ≤ q ≤ p ∗ , with 1 p ∗ = 1 p − kn , (A.5) W k + d,p ( U ) ֒ → W d,q ( U ) if kp = n and p ≤ q < ∞ . This last inclusion can then be composed with (A.2) for an arbitrarily large choiceof q , giving another inclusion(A.6) W k + d,p ( U ) ֒ → C d − ,α ( U ) if kp = n and 0 < α < . Remark
A.7 . The embedding theorem suggests that one should intuitively thinkof W k,p ( U ) as consisting of functions with “ k − n/p continuous derivatives,” wherethe number k − n/p may in general be a non-integer and/or negative. This providesa useful mnemonic for results about embeddings of one Sobolev space into another,such as the following. ectures on Symplectic Field Theory Corollary
A.8 . Assume
U ⊂ R n is an open subset satisfying the strong localLipschitz condition, ≤ p, q < ∞ , and k, m ≥ are integers satisfying k ≥ m, p ≤ q, and k − np ≥ m − nq . Then there exists a continuous inclusion W k,p ( U ) ֒ → W m,q ( U ) . (cid:3) By the Arzel`a-Ascoli theorem, the natural inclusion C k,α ′ ( U ) ֒ → C k,α ( U )for α < α ′ is a compact operator whenever U ⊂ R n is bounded. It follows that if U ⊂ R n in (A.2) is bounded and α is strictly less than the extremal value k − n/p , thenthe inclusion (A.2) is also compact. A similar statement holds for the inclusion (A.4)when p ≤ q < p ∗ , and this is known as the Rellich-Kondrachov compactnesstheorem . We summarize these as follows:
Theorem
A.9 ([
AF03 , § . Assume
U ⊂ R n is a bounded Lipschitz domain, k ≥ and d ≥ are integers and ≤ p < ∞ .(1) If kp > n and k − n/p < , then the inclusions W k + d,p ( U ) ֒ → C d,α ( U ) for α ∈ (0 , k − n/p ) ,W k + d,p ( U ) ֒ → W d,q ( U ) for q ∈ [ p, ∞ ) are compact.(2) If kp ≤ n and p ∗ ∈ ( p, ∞ ] is defined by the condition /p ∗ = 1 /p − k/n ,then the inclusions W k + d,p ( U ) ֒ → W d,q ( U ) for q ∈ [ p, p ∗ ) are compact.In particular, the continuous inclusion W k,p ( U ) ֒ → W m,q ( U ) in Corollary A.8 iscompact whenever the inequality k − n/p ≥ m − n/q is strict. (cid:3) A.2. Products, compositions, and rescaling
We now restate and prove Propositions 2.7, 2.8 and 2.10 from § U ⊂ R n , and the restrictions on U can be dropped at the costof replacing each space W k,p by W k,p .We begin by generalizing Prop. 2.7, hence we consider Sobolev spaces of functionsvalued in R or C so that pointwise products of functions are well defined almosteverywhere. We say that there is a continuous product map , W k ,p ( U ) × . . . × W k m ,p m ( U ) → W k,p ( U ) , or a continuous product pairing in the case m = 2, if for every set of functions f i ∈ W k i ,p i ( U ) with i = 1 , . . . , m , the pointwise product function f · . . . · f m is in W k,p ( U ) and there is an estimate of the form k f · . . . · f m k W k,p ≤ c k f k W k ,p · . . . · k f m k W km,pm Chris Wendl for some constant c > f , . . . , f m . The case m = 2, k = k = k and p = p = p is especially interesting, as the space W k,p ( U ) is then a Banachalgebra . More generally, one can ask under what circumstances multiplication byfunctions of class W k,p defines a bounded linear operator on functions of class W m,q .A hint about this comes from the world of classically differentiable functions: mul-tiplication by C k -smooth functions defines a continuous map C m → C m if and onlyif k ≥ m . The corresponding answer in Sobolev spaces turns out to be that func-tions of class W k,p need to have strictly more than zero derivatives in the sense ofRemark A.7, and at least as many derivatives as functions of class W m,q . Theorem
A.10 . Assume
U ⊂ R n is an open subset satisfying the strong localLipschitz condition, k , p , m and q satisfy the same numerical hypotheses as inCorollary A.8 (so in particular W k,p ( U ) embeds continuously into W m,q ( U ) ), and kp > n . Then there exists a continuous product pairing W k,p ( U , C ) × W m,q ( U , C ) → W m,q ( U , C ) : ( f, g ) f g. The following preparatory lemma will be useful both for proving the productestimate and for further results below. It is an easy consequence of Theorem A.6and H¨older’s inequality.
Lemma
A.11 . Assume
U ⊂ R n is an open subset satisfying the strong lo-cal Lipschitz condition, m ≥ is an integer, and we are given positive numbers p , . . . , p m ≥ and integers k , . . . , k m ≥ . Let I := (cid:8) i ∈ { , . . . , m } (cid:12)(cid:12) k i p i ≤ n (cid:9) .Then for any q ≥ satisfying X i ∈ I (cid:18) p i − k i n (cid:19) < q ≤ m X i =1 p i , there is a continuous product map W k ,p ( U ) × . . . × W k m ,p m ( U ) → L q ( U ) . Proof.
By the generalized H¨older inequality (A.1), it suffices to show that forany q ≥ q , . . . , q m ∈ [ q, ∞ ] satisfying1 /q = 1 /q + . . . + 1 /q m for which Theorem A.6 provides continuous inclusions W k i ,p i ( U ) ֒ → L q i ( U )for each i = 1 , . . . , m . Whenever k i p i > n , this inclusion is valid with q i chosen freelyfrom the interval [ p i , ∞ ], so 1 /q i can then take any value subject to the constraint0 ≤ q i ≤ p i . If on the other hand k i p i ≤ n , then we can arrange 1 /q i to take any value in therange 1 p i − k i n < q i ≤ p i . Adding these up, the range of values for P i q i that we can achieve in this way coversthe stated interval. (cid:3) ectures on Symplectic Field Theory Proof of Theorem A.10.
By density of smooth functions, it suffices to provethat an estimate of the form k f g k W m,q ≤ c k f k W k,p k g k W m,q holds for all f ∈ C ∞ ( U ) ∩ W k,p ( U ) and g ∈ C ∞ ( U ) ∩ W m,q ( U ). Equivalently, weneed to show that for all f and g of this type and every multiindex α of degree | α | ≤ m , there is a constant c > f and g such that k ∂ α ( f g ) k L q ≤ c k f k W k,p k g k W m,q . Since f and g are smooth, we are free to use the product rule in computing ∂ α ( f g ),which will then be a linear combination of terms of the form ∂ β f · ∂ γ g where | α | = | β | + | γ | , hence we have reduced the problem to proving a bound k ∂ β f · ∂ γ g k L q ≤ c k f k W k,p k g k W m,q for every pair of multiindices β , γ with | β | + | γ | ≤ m . Since ∂ β f ∈ W k −| β | ,p ( U ) and ∂ γ f ∈ W m −| γ | ,q ( U ), the result follows if we can assume that for every pair of integers a, b ≥ a + b ≤ m , there exists a continuous product pairing(A.7) W k − a,p ( U ) × W m − b,q ( U ) → L q ( U ) . If ( k − a ) p > n , then W k − a,p ֒ → L ∞ and (A.7) is immediate since W m − b,q ֒ → L q ( U ).For the remaining cases, we shall apply Lemma A.11, noting that the condition1 /q ≤ /p + 1 /q is trivially satisfied.If ( m − b ) q > n but ( k − a ) p ≤ n , then the hypotheses of the lemma are satisfiedif and only if 1 p − k − an < q . Since p − kn ≤ q − mn by assumption, we have1 p − k − an = 1 p − kn + an ≤ q − mn + an ≤ q since a ≤ m , and equality holds only if a = m , b = 0 and k − n/p = m − n/q ,which implies mq > n . In this case W m − b,q = W m,q ֒ → L ∞ , and the pairing (A.7)follows because W k − a,p = W k − m,p embeds continuously into L q : the latter followsfrom Theorem A.6 since p − k − mn = q .Finally, when ( k − a ) p ≤ n and ( m − b ) q ≤ n , the hypotheses of the lemma aresatisfied since (cid:18) p − k − an (cid:19) + (cid:18) q − m − bn (cid:19) ≤ p − kn + 1 q − mn + mn = (cid:18) p − kn (cid:19) + 1 q < q , where we’ve used the assumption kp > n and the fact that a + b ≤ m . (cid:3) The next result generalizes Proposition 2.8 and concerns the following question:if f : U → R m is a function of class W k,p whose graph lies in some open subset V ⊂ U × R m , and Ψ : V → R N is another function, under what conditions can weconclude that the function U → R N : x Ψ( x, f ( x )) Chris Wendl is in W k,p ( U , R N )? We will abbreviate this function in the following by Ψ ◦ (Id × f ),and we would also like to know whether it depends continuously (in the W k,p -topology) on f and Ψ. The following theorem is stated rather generally, but on firstreading you may prefer to assume U ⊂ R n is bounded, in which case some of thehypotheses become vacuous. We will say that an open subset V ⊂ U × R m is a star-shaped neighborhood of f : U → R m if it contains the graph of f and( x, v ) ∈ V ⇒ ( x, tv + (1 − t ) f ( x )) ∈ V for all t ∈ [0 , . Theorem
A.12 . Assume
U ⊂ R n is an open subset satisfying the strong localLipschitz condition, p ∈ [1 , ∞ ) and k ∈ N satisfy kp > n , and V ⊂ U × R m is a star-shaped neighborhood of some function f ∈ W k,p ( U , R m ) . Assume also O k,p ( U ; V ) ⊂ W k,p ( U , R m ) is an open neighborhood of f such that ( x, f ( x )) ∈ V for all x ∈ U and f ∈ O k,p ( U ; V ) , and O k ( V , R N ) ⊂ C k ( V , R N ) is a subset such that all Ψ ∈ O k ( V , R n ) have thefollowing properties: (1) There exists a bounded subset K ⊂ U such that Ψ( x, v ) is independent of x for all x ∈ U \ K ;(2) Ψ ◦ (Id × f ) ∈ L p ( U , R N ) .Then there is a well-defined and continuous map O k ( V , R N ) × O k,p ( U ; V ) → W k,p ( U , R N ) : (Ψ , f ) Ψ ◦ (Id × f ) . Proof.
We will show first that if f ∈ O k,p ( U ; V ) is smooth, then Ψ ◦ (Id × f ) belongs to W k,p ( U , R N ) for every Ψ ∈ O k ( V , R N ). Since V is a star-shapedneighborhood of f , we have | Ψ( x, f ( x )) − Ψ( x, f ( x )) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ddt Ψ (cid:0) x, tf ( x ) + (1 − t ) f ( x ) (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z | D Ψ (cid:0) x, tf ( x ) + (1 − t ) f ( x ) (cid:1) | dt (cid:19) · | f ( x ) − f ( x ) |≤ k Ψ k C ( V ) · | f ( x ) − f ( x ) | for all x ∈ U , implying k Ψ ◦ (Id × f ) − Ψ ◦ (Id × f ) k L p ≤ k Ψ k C ( V ) · k f − f k L p , hence Ψ ◦ (Id × f ) ∈ L p ( U , R N ).For ℓ = 1 , . . . , k , we can regard the ℓ th derivative of Ψ with respect to variablesin R m as a bounded and uniformly continuous map from V into the vector space ofsymmetric ℓ -multilinear maps from R m to R N , denoting this by D ℓ Ψ :
V →
Hom(( R m ) ⊗ ℓ , R N ) . Denote the partial derivatives with respect to variables in
U ⊂ R n by D β Ψ :
V → R N , Both of the conditions on Ψ ∈ O k ( V , R n ) are vacuous if U ⊂ R n is bounded. ectures on Symplectic Field Theory where β is a multiindex in n variables. Now for any multiindex α with | α | ≤ k , thederivative ∂ α (Ψ ◦ (Id × f )) is a linear combination of product functions of the form(A.8) ( D γ D ℓ Ψ ◦ (Id × f ))( ∂ β f, . . . , ∂ β ℓ f ) : U → R N , where ℓ + | γ | ∈ { , . . . , | α |} and | β | + . . . + | β ℓ | = | α | − | γ | . If ℓ = 0 but | γ | > L p ( U , R N ) since it is continuous and D γ Ψ( x, v ) = 0for x ∈ U \ K , where K is bounded. For ℓ ≥
1, it satisfies (cid:13)(cid:13) ( D γ D ℓ Ψ ◦ (Id × f ))( ∂ β f, . . . , ∂ β ℓ f ) (cid:13)(cid:13) L p ( U ) ≤ k D γ D ℓ Ψ k C ( V ) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ Y j =1 | ∂ β j f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( U ) if the product on the right hand side has finite L p -norm. The latter is triviallytrue if ℓ = 1. To deal with the ℓ ≥ ∂ β j f ∈ W k −| β j | ,p ( U ) for each j = 1 , . . . , ℓ , so the necessary bound will follow from the existence of a continuousproduct map W k − m ,p ( U ) × . . . × W k − m ℓ ,p ( U ) → L p ( U )for m j := | β j | , and we claim that such a product map does exist whenever kp > n and m , . . . , m ℓ ≥ m + . . . + m ℓ ≤ k . To see this, notefirst that since W k − m j ,p ֒ → L ∞ whenever ( k − m j ) p > n , it suffices to prove theclaim under the assumption that ( k − m j ) p ≤ n for every j = 1 , . . . , ℓ . In this case,Lemma A.11 provides the desired product map if the condition ℓ X j =1 (cid:18) p − k − m j n (cid:19) < p ≤ ℓ X j =1 p is satisfied. And it is: using kp > n , ℓ ≥ m + . . . + m ℓ ≤ k , we find ℓ X j =1 (cid:18) p − k − m j n (cid:19) = ℓ (cid:18) p − kn (cid:19) + m + . . . + m ℓ n ≤ p + ( ℓ − (cid:18) p − kn (cid:19) < p . This proves that Ψ ◦ (Id × f ) ∈ W k,p ( U , R N ).Next, suppose f ∈ O k,p ( U ; V ) is not necessarily smooth but f i ∈ O k,p ( U ; V ) isa sequence of smooth functions converging to f in W k,p , while Ψ i ∈ O k ( V , R N )converges to Ψ ∈ O k ( V , R N ) in C k . Then the same argument we used to estimate k Ψ ◦ (Id × f ) − Ψ ◦ (Id × f ) k L p shows that Ψ ◦ (Id × f i ) → Ψ ◦ (Id × f ) in L p , andsince f i is also C -convergent, the compactly supported functions D γ Ψ ◦ (Id × f i )converge to D γ Ψ ◦ (Id × f ) in L p for each multiindex with 1 ≤ | γ | ≤ k . For ℓ ≥ | γ | + ℓ ≤ k , D γ D ℓ Ψ i ◦ (Id × f i ) converges to D γ D ℓ Ψ ◦ (Id × f ) in C ( U , R N ),and each of the derivatives ∂ β j f i appearing in (A.8) also converges in L p ( U ). Inlight of the continuous product maps discussed above, it follows that each derivative ∂ α (Ψ i ◦ (Id × f i )) for | α | ≤ k is L p -convergent, and its limit is necessarily thecorresponding weak derivative ∂ α (Ψ ◦ (Id × f )), hence (see Exercise A.13 below)Ψ ◦ (Id × f ) ∈ W k,p ( U , R N ) and Ψ i ◦ (Id × f i ) W k,p −→ Ψ ◦ (Id × f ). (cid:3) Chris Wendl
Exercise
A.13 . Show that if f i is a sequence of smooth functions on an open set U ⊂ R n with f i L p → f and ∂ α f i L p → g for some multiindex α and functions f, g ∈ L p ( U ),then ∂ α f = g in the sense of distributions.The following result on coordinate transformations of the domain can be provedin an analogous way to Theorem A.12, though it is considerably easier since there isno need to worry about Sobolev product maps (and thus no need to assume kp > n or impose regularity conditions on the domain). Theorem
A.14 ([
AF03 , § . Assume k ∈ N , ≤ p ≤ ∞ , and U , U ′ ⊂ R n are open subsets with a C k -smooth diffeomorphism ϕ : U → U ′ such that allderivatives of ϕ and ϕ − up to order k are bounded and uniformly continuous. Thenthere is a well-defined Banach space isomorphism W k,p ( U ′ ) → W k,p ( U ) : f f ◦ ϕ. (cid:3) We now restate and prove Proposition 2.10. We denote by ˚ D n , ˚ D nǫ ⊂ R n the openballs of radius 1 and ǫ respectively about the origin. Theorem
A.15 . Assume p ∈ [1 , ∞ ) and k ∈ N satisfy kp > n , and for each f ∈ W k,p (˚ D n ) and ǫ ∈ (0 , , define f ǫ ∈ W k,p (˚ D n ) by f ǫ ( x ) := f ( ǫx ) . Then there exist constants
C > and r > such that for every f ∈ W k,p (˚ D n ) , k f ǫ − f (0) k W k,p (˚ D n ) ≤ Cǫ r k f − f (0) k W k,p (˚ D n ) for all ǫ ∈ (0 , . Proof.
Let β denote a multiindex of order | β | = k . Then using a change ofvariables, we have k ∂ β ( f ǫ − f (0)) k pL p (˚ D n ) = ǫ kp Z D n | ∂ β f ( ǫx ) | p = ǫ kp − n Z D nǫ | ∂ β f ( x ) | p ≤ ǫ kp − n k ∂ β f k pL p (˚ D n ) ≤ ǫ kp − n k f − f (0) k pW k,p (˚ D n ) , and ǫ kp − n → ǫ → kp − n > | β | = m ∈ { , . . . , k − } . Then ∂ β f and ∂ β f ǫ are in W k − m,p (˚ D n ),and if ( k − m ) p < n , Theorem A.6 gives a continuous inclusion(A.9) W k − m,p (˚ D n ) ֒ → L q (˚ D n )with q > p satisfying 1 /q + ( k − m ) /n = 1 /p . Likewise, if ( k − m ) p ≥ n , then (A.9)is a continuous inclusion for arbitrarily large choices of q ≥ p . We will thereforeassume in general that (A.9) holds with q ∈ ( p, ∞ ) satisfying1 q + 1 r = 1 p , ectures on Symplectic Field Theory where r = nk − m if ( k − m ) p < n and otherwise r = p + δ for some δ > k ∂ β ( f ǫ − f (0)) k pL p (˚ D n ) = ǫ mp Z D n | ∂ β f ( ǫx ) | p = ǫ mp − n Z D nǫ | ∂ β f ( x ) | p ≤ ǫ mp − n k ∂ β f k pL q (˚ D nǫ ) k k pL r (˚ D nǫ ) ≤ ǫ mp − n [Vol( D nǫ )] p/r k ∂ β f k pL q (˚ D n ) ≤ cǫ mp − n [Vol( D nǫ )] p/r k ∂ β f k pW k − m,p (˚ D n ) ≤ cǫ mp − n [Vol( D nǫ )] p/r k f − f (0) k pW k,p (˚ D n ) for some constant c >
0. Writing Vol( D nǫ ) = Cǫ n for a suitable constant C >
0, theexponent on ǫ in this expression becomes mp − n + npr , which is positive whenever r = p + δ with δ > m ≥
1, andin the case r = n/ ( k − m ), it becomes simply kp − n > L p -norm of f ǫ − f (0) itself, we can use the fact that f ∈ W k,p is H¨older continuous, i.e. it satisfies | f ( x ) − f (0) | ≤ c k f − f (0) k W k,p (˚ D n ) | x | α for all x ∈ ˚ D n for suitable constants c > α ∈ (0 , k f ǫ − f (0) k pL p (˚ D n ) = Z D n | f ( ǫx ) − f (0) | p ≤ c p k f − f (0) k pW k,p Z D n | ǫx | αp = c p k f − f (0) k pW k,p ǫ αp Z D n | x | αp = ǫ αp c p Vol( S n − ) αp + n k f − f (0) k pW k,p . (cid:3) A.3. Spaces of sections of vector bundles
In this section, fix a field F := R or C , assume M is a smooth n -dimensional manifold, possibly with boundary, and π : E → M is a smooth vector bundle of rank m over F . This comes with a “bundleatlas” A ( π ), a set whose elements α ∈ A ( π ) each consist of the following data:(1) An open subset U α ⊂ M ;(2) A smooth local coordinate chart ϕ α : U α ∼ = −→ Ω α , where Ω α is an opensubset of R n + := { ( x , . . . , x n ) ∈ R n | x n ≥ } ;(3) A smooth local trivialization Φ α : E | U α ∼ = −→ U α × F m . Chris Wendl
Smoothness of ϕ α and Φ α means as usual that for every pair α, β ∈ A ( π ), thecoordinate transformations ϕ βα := ϕ − β ◦ ϕ α : Ω αβ ∼ = −→ Ω βα , Ω αβ := ϕ α ( U α ∩ U β )and transition maps g βα : U α ∩ U β → GL( m, F ) such that Φ β ◦ Φ − α ( x, v ) = ( x, g βα ( x ) v )for x ∈ U α ∩ U β , v ∈ F m are smooth, and we shall assume the bundle atlas is maximal in the sense thatany triple ( U , ϕ, Φ) that is smoothly compatible with every α ∈ A ( π ) also belongsto A ( π ).Any α ∈ A ( π ) now associates to sections η : M → E their local coordinaterepresentatives η α := pr ◦ Φ α ◦ η ◦ ϕ − α : Ω α → F m , where pr : U α × F m → F m is the projection, and the representatives with respectto two distinct α, β ∈ A ( π ) are related by η β = ( g βα ◦ ϕ − β )( η α ◦ ϕ αβ ) on Ω βα ⊂ Ω β . For p ∈ [1 , ∞ ] and each integer k ≥
0, we then define the topological vector spaceof sections of class W k,p loc by W k,p loc ( E ) := (cid:8) η : M → E (cid:12)(cid:12) sections such that η α ∈ W k,p loc (˚Ω α , F m )for all α ∈ A ( π ) (cid:9) , where convergence η j → η in W k,p loc ( E ) means that η αj → η α in W k,p loc (˚Ω α , F m ) for all α ∈ A ( π ). Note that Ω α is not necessarily an open subset of R n since it may containpoints in ∂ R n + = R n − × { } , but its interior ˚Ω α is open in R n , and W k,p loc (˚Ω α ) isthus defined as in § A.1. Strictly speaking, elements of η ∈ W k,p loc ( E ) are not sectionsbut equivalence classes of sections defined almost everywhere—the latter notion isdefined with respect to any measure arising from a smooth volume element on M ,and it does not depend on this choice.It turns out that W k,p loc ( E ) can be given the structure of a Banach space if M iscompact. This follows from the fact that M can then be covered by a finite subset ofthe atlas A ( π ), but we must be a little bit careful: not all charts in A ( π ) are equallysuitable for defining W k,p -norms on sections, because e.g. even a nice smooth section η ∈ Γ( E ) may have k η α k W k,p (˚Ω α ) = ∞ if Ω α ⊂ R n + is unbounded. One way to dealwith this is as follows: we will say that α ∈ A ( π ) is a precompact chart if thereexists α ′ ∈ A ( π ) and a compact subset K ⊂ M such that U α ⊂ K ⊂ U α ′ . When this is the case, Ω α ⊂ R n + is necessarily bounded, and the transition mapsbetween two precompact charts necessarily have bounded derivatives of all orders,as they are restrictions to precompact subsets of maps that are smooth on largerdomains. If M is compact, then one can always find a finite subset I ⊂ A ( π )consisting of precompact charts such that M = S α ∈ I U α . ectures on Symplectic Field Theory Definition
A.16 . Suppose E → M is a smooth vector bundle over a compactmanifold M , and I ⊂ A ( π ) is a finite set of precompact charts such that {U α } α ∈ I is an open cover of M . We then define W k,p ( E ) as the vector space of all sections η : M → E for which the norm k η k W k,p := k η k W k,p ( E ) := X α ∈ I k η α k W k,p (˚Ω α ) is finite.The norm in the above definition depends on auxiliary choices, but it is easy tosee that the resulting definition of the space W k,p ( E ) and its topology do not. Infact: Proposition
A.17 . If M is compact, then W k,p ( E ) = W k,p loc ( E ) , and a sequence η j converges to η in W k,p loc ( E ) if and only if the norm given in Definition A.16 satisfies k η j − η k W k,p ( E ) → . The proposition is an immediate consequence of the following.
Lemma
A.18 . Suppose M is a smooth manifold, π : E → M is a smooth vectorbundle, { β } ∪ J ⊂ A ( π ) is a finite collection of charts such that M = S α ∈ J U α andall coordinate transformations and transition maps relating any two charts in thecollection { β } ∪ J have bounded derivatives of all orders (e.g. it suffices to assumeall are precompact). Then there exists a constant c > such that k η β k W k,p (˚Ω β ) ≤ c X α ∈ J k η α k W k,p (˚Ω α ) for all sections η : M → E with η α ∈ W k,p (˚Ω α ) for every α ∈ J . Proof.
Choose a partition of unity { ρ α : M → [0 , } α ∈ J subordinate to thefinite open cover {U α } α ∈ J . Now η = P α ∈ J ρ α η , and each ρ α η is supported in U α , so( ρ α η ) β has support in Ω βα = ϕ β ( U α ∩ U β ). Thus using Theorem A.14 with the factthat g βα , ϕ − β , ϕ αβ and ϕ βα = ϕ − αβ are all smooth functions with bounded derivativesof all orders on the domains in question, we find k η β k W k,p (˚Ω β ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ J ( ρ α η ) β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W k,p (˚Ω β ) ≤ X α ∈ J k ( ρ α η ) β k W k,p (˚Ω βα ) = X α ∈ J k ( ρ α ◦ ϕ − β )( g βα ◦ ϕ − β )( η α ◦ ϕ αβ ) k W k,p (˚Ω βα ) ≤ c X α ∈ J k η α k W k,p (˚Ω αβ ) ≤ c X α ∈ J k η α k W k,p (˚Ω α ) . (cid:3) Corollary
A.19 . If M is compact, then the norm on W k,p ( E ) given by Defi-nition A.16 is independent of all auxiliary choices up to equivalence of norms. (cid:3) Theorem
A.20 . For any smooth vector bundle π : E → M over a compactmanifold M , W k,p ( E ) is a Banach space. Chris Wendl
Proof. If η j ∈ W k,p ( E ) is a Cauchy sequence, then for some chosen finitecollection I ⊂ A ( π ) of precompact charts covering M , the sequences η αj for α ∈ I areCauchy in W k,p (˚Ω α ) and thus have limits ξ ( α ) ∈ W k,p (˚Ω α , F m ). Choosing a partitionof unity { ρ α : M → [0 , } α ∈ I subordinate to {U α } α ∈ I , we can now associate to each α ∈ I a section η ∞ ,α ∈ W k,p ( E ) characterized uniquely by the condition that itvanishes outside of U α and is represented in the trivialization on U α by η α ∞ ,α = ( ρ α ◦ ϕ − α ) ξ ( α ) . We claim that ρ α η j → η ∞ ,α in W k,p ( E ) for each α ∈ I . Indeed, we have( ρ α η j ) α = ( ρ α ◦ ϕ − α ) η αj → ( ρ α ◦ ϕ − α ) ξ ( α ) = η α ∞ ,α in W k,p (˚Ω α )since η αj → ξ ( α ) . For all other β ∈ I not equal to α , ( ρ α η j ) β − η β ∞ ,α ∈ W k,p (˚Ω β , F m )has support in Ω βα = ϕ β ( U α ∩ U β ), thus k ( ρ α η j ) β − η β ∞ ,α k W k,p (˚Ω β ) = k ( ρ α η j ) β − η β ∞ ,α k W k,p (˚Ω βα ) ≤ c k ( ρ α η j ) α − η α ∞ ,α k W k,p (˚Ω α ) , where the inequality comes from Lemma A.18 after replacing M with U α , and U β with U β ∩ U α (note that the lemma does not require M to be compact). With theclaim established, we have η j = X α ∈ I ρ α η j → X α ∈ I η ∞ ,α in W k,p ( E ) . (cid:3) Exercise
A.21 . For
U ⊂ R n an open subset, the space W k,p loc ( U ) was defined in § A.1, but one can give it an alternative definition in the present context by viewingfunctions on U as sections of a trivial vector bundle over U , with the latter viewedas a noncompact smooth n -manifold. Show that these two definitions of W k,p loc ( U )are equivalent. Exercise
A.22 . Suppose
U ⊂ R n is a bounded open subset with smooth bound-ary, so its closure U ⊂ R n is a smooth compact submanifold with boundary, and let E → U be a trivial vector bundle. Show that there is a canonical Banach space iso-morphism between W k,p ( U ) as defined in § A.1 and W k,p ( E ) as defined in the presentsection. Hint: Recall that sections in W k,p ( E ) are only required to be defined almosteverywhere, so in particular if the domain M is a manifold with boundary, they neednot be well defined on ∂M . In light of Exercise A.22, the natural generalization of W k,p ( U ) in the presentsetting is W k,p ( E ) := C ∞ ( E | M \ ∂M ) , i.e. it is the closure in the W k,p -norm of the space of smooth sections that vanishnear the boundary. Density of smooth sections will imply that this is the same as W k,p ( E ) if M is closed, but in general W k,p ( E ) is a closed subspace of W k,p ( E ).The partition of unity argument in Theorem A.20 contains all the essential ideasneeded to generalize results about Sobolev spaces on domains in R n to compactmanifolds. We now state the essential results, leaving the proofs as exercises. ectures on Symplectic Field Theory Theorem
A.23 . Assume M is a smooth compact n -manifold, possibly withboundary, π : E → M is a smooth vector bundle of finite rank, k ≥ is an in-teger and ≤ p < ∞ . Then the Banach space W k,p ( E ) has the following properties.(1) The space Γ( E ) of smooth sections is dense in W k,p ( E ) .(2) If N ⊂ M is a smooth compact n -dimensional submanifold with boundary,then there exists a bounded linear operator E : W k,p ( E | N ) → W k,p ( E ) which is an extension operator in the sense that Eη | N = η for all η ∈ W k,p ( E | N ) . Moreover, a section η ∈ W k,p ( E | N ) belongs to W k,p ( E | N ) ifand only if the section ˜ η defined to match η on N and to vanish on M \ N belongs to W k,p ( E ) .(3) If kp > n , then for each integer d ≥ , there exists a continuous and compactinclusion W k + d,p ( E ) ֒ → C d ( E ) . (4) The natural inclusion W k +1 ,p ( E ) ֒ → W k,p ( E ) is compact.(5) Suppose F, G → M are smooth vector bundles such that there exists asmooth bundle map E ⊗ F → G : η ⊗ ξ η · ξ. Then if kp > n and ≤ m ≤ k , there exists a continuous product pairing W k,p ( E ) × W m,p ( F ) → W m,p ( G ) : ( η, ξ ) η · ξ. In particular, products of W k,p sections give W k,p sections whenever kp > n .(6) Suppose F → M is another smooth vector bundle, V ⊂ E is an open subsetthat intersects every fiber of E , and we consider the spaces W k,p ( V ) := (cid:8) η ∈ W k,p ( E ) (cid:12)(cid:12) η ( M ) ⊂ V (cid:9) and C kM ( V , F ) := (cid:8) Φ :
V → F | fiber-preserving maps of class C k (cid:9) , where the latter is assigned the topology of C k -convergence on compact sub-sets. If kp > n , then W k,p ( V ) is an open subset of W k,p ( E ) , and the map C kM ( V , F ) × W k,p ( V ) → W k,p ( F ) : (Φ , η ) Φ ◦ η is well defined and continuous.(7) If N is another smooth compact manifold and ϕ : N → M is a smoothdiffeomorphism, then there is a Banach space isomorphism W k,p ( E ) → W k,p ( ϕ ∗ E ) : η η ◦ ϕ. (cid:3) Chris Wendl
A.4. Some remarks on domains with cylindrical ends
For bundles π : E → M with M noncompact, W k,p ( E ) is not generally welldefined without making additional choices. When M = ˙Σ = Σ \ Γ is a puncturedRiemann surface and π : E → ˙Σ is equipped with an asymptotically Hermitian struc-ture { ( E z , J z , ω z ) } z ∈ Γ , one nice way to define W k,p ( E ) was introduced in § W k,p loc ( E ) whose W k,p -norms on each cylindricalend are finite with respect to a choice of asymptotic trivialization. This definitionrequires the convenient fact that complex vector bundles over S are always triv-ial, though one can also do without this by using the ideas in the previous section.Indeed, any collection of local trivializations on the asymptotic bundle E z → S covering S gives rise via the asymptotically Hermitian structure to a collection oftrivializations on E covering the corresponding cylindrical end ˙ U z . The key factis then that S is compact, hence one can always choose such a covering to be fi-nite: combining this with a finite covering of ˙Σ in the complement of its cylindricalends by precompact charts, we obtain a covering of ˙Σ by a finite collection of bun-dle charts that are not all precompact, but nonetheless have the property that alltransition maps have bounded derivatives of all orders. This is enough to define a W k,p -norm for sections of E → ˙Σ as in Definition A.16 and to prove that it doesnot depend on the choices of charts or local trivializations, though it does dependon the asymptotically Hermitian structure.With this definition understood, one can easily generalize the Sobolev embeddingtheorem and other important statements in Theorem A.23 to the setting of anasymptotically Hermitian bundle over a punctured Riemann surface. We shall leavethe details of this generalization as an exercise, but take the opportunity to pointout a few important differences from the compact case.First, since ˙Σ is not compact, neither are the inclusions W k + d,p ( E ) ֒ → C d ( E ) , W k +1 ,p ( E ) ֒ → W k,p ( E ) . The proof of compactness fails due to the fact that cylindrical ends require localtrivializations over unbounded domains of the form (0 , ∞ ) × (0 , ⊂ R , for whichTheorem A.9 does not hold. And indeed, considering unbounded shifts on theinfinite cylinder ˙Σ = R × S , it is easy to find a sequence of W k,p -bounded functionswith kp > C -convergent subsequence. That is the bad news.The good news is that if η ∈ W k + d,p ( E ) for kp >
2, then one can say considerablymore about η than just that it is C d -smooth. Indeed, restricting to one of thecylindrical ends [0 , ∞ ) × S ⊂ ˙Σ, notice that finiteness of the W k + d,p -norm over ˙Σimplies k η k W k + d,p (( R, ∞ ) × S ) → R → ∞ . Since these domains are all naturally diffeomorphic for different values of R , the C d -norm of η over ( R, ∞ ) × S is bounded by the W k + d,p -norm via a constant thatdoes not depend on R , so this implies an asymptotic decay condition k η k C d ([ R, ∞ ) × S ) → R → ∞ for every η ∈ W k + d,p ( E ). ectures on Symplectic Field Theory Here is another useful piece of good news: since ˙Σ does not have boundary, W k,p ( E ) = W k,p ( E ). Theorem
A.24 . Given an asymptotically Hermitian bundle E over a puncturedRiemann surface ˙Σ , the space C ∞ ( E ) of smooth sections with compact support isdense in W k,p ( E ) for all k ≥ and ≤ p < ∞ . Proof.
We can assume as in Definition A.16 that the W k,p -norm for sections η of E is given by k η k W k,p = X α ∈ I k η α k W k,p (Ω α ) , where I ⊂ A ( π ) is a finite collection of bundle charts α = (cid:16) ϕ α : U α ∼ = −→ Ω α , Φ α : E | U α ∼ = −→ U α × C n (cid:17) such that each of the open sets Ω α ⊂ C is either bounded or (for charts over thecylindrical ends) of the formΩ α = (0 , ∞ ) × ω α ⊂ R = C for some bounded open subset ω α ⊂ R . Now given η ∈ W k,p ( E ), Theorem A.1provides for each α ∈ I a sequence η αj ∈ W k,p (Ω α ) of smooth functions with boundedsupport such that η αj → η α in W k,p (Ω α ). Choose a partition of unity { ρ α : ˙Σ → [0 , } α ∈ I subordinate to the open cover {U α } α ∈ I and let η j := X α ∈ I ρ α ( η αj ◦ ϕ α ) ∈ W k,p ( E ) . These sections are smooth and have compact support since the η αj have boundedsupport in Ω α , and they converge in W k,p to η . (cid:3) PPENDIX B
The Floer C ε space The C ε -topology for functions was introduced by Floer [ Flo88b ] to provide a Ba-nach manifold of perturbed geometric structures without departing from the smoothcategory: it is a way to circumvent the annoying fact that spaces of smooth functionswhich arise naturally in geometric settings are not Banach spaces. The constructionof C ε spaces generally depends on several arbitrary choices and is thus far fromcanonical, but this detail is unimportant since the C ε space itself is never the mainobject of interest. What is important is merely the properties that it has, namelythat it not only embeds continuously into C ∞ and contains an abundance of non-trivial functions, but also is a separable Banach space and can therefore be used inthe Sard-Smale theorem for genericity arguments. We shall prove these facts in thisappendix.Fix a smooth finite-rank vector bundle π : E → M over a finite-dimensionalcompact manifold M , possibly with boundary. For each integer k ≥
0, we denote by C k ( E ) the Banach space of C k -smooth sections of E ; note that the norm on C k ( E )depends on various auxiliary choices but is well defined up to equivalence of normssince M is compact. Now if ( ε k ) ∞ k =0 is a sequence of positive numbers with ε k → C ε ( E ) = (cid:8) η ∈ Γ( E ) (cid:12)(cid:12) k η k C ε < ∞ (cid:9) , where the C ε -norm is defined by(B.1) k η k C ε = ∞ X k =0 ε k k η k C k . The norm for C ε ( E ) is somewhat more delicate than for C k ( E ), e.g. its equivalenceclass is not obviously independent of auxiliary choices. This remark is meant asa sanity check, but it should not cause extra concern since, in practice, the space C ε ( E ) is typically regarded as an auxiliary choice in itself. In many applications,one fixes an open subset U ⊂ M and considers the closed subspace C ε ( E ; U ) = (cid:8) η ∈ C ε ( E ) (cid:12)(cid:12) η | M \U ≡ (cid:9) . Remark
B.1 . The requirement for M to be compact can be relaxed as long as U ⊂ M has compact closure: e.g. in one situation of frequent interest in this book,we take M to be the noncompact completion of a symplectic cobordism. In this case C ε ( E ; U ) can be defined as a closed subspace of C ε ( E | M ) where M ⊂ M is anycompact manifold with boundary that contains the closure of U . For this reason,we lose no generality in continuing under the assumption that M is compact. Chris Wendl
In order to prove things about C ε ( E ), we will need to specify a more precisedefinition of the C k -norms. To this end, define a sequence of vector bundles E ( k ) → M for integers k ≥ E (0) := E, E ( k +1) := Hom( T M, E ( k ) ) . Choose connections and bundle metrics on both
T M and E ; these induce connec-tions and bundle metrics on each of the E ( k ) , so that for any section ξ ∈ Γ( E ( k ) ),the covariant derivative ∇ ξ is now a section of E ( k +1) . In particular for η ∈ Γ( E ),we can define the “ k th covariant derivative” of η as a section ∇ k η ∈ Γ( E ( k ) ) . Using the bundle metrics to define C -norms for sections of E ( k ) , we can then define k η k C k ( E ) = k X m =0 k∇ m η k C ( E ( m ) ) , where by convention ∇ η := η . We will assume throughout the following that the C k -norms appearing in (B.1) are defined in this way. Theorem
B.2 . C ε ( E ) is a Banach space. Proof.
We need to show that C ε -Cauchy sequences converge in the C ε -norm.It is clear from the definitions that if η j ∈ C ε ( E ) is Cauchy, then η j is also C k -Cauchy for every k ≥
0, hence its derivatives ∇ k η j for every k are C -convergentto continuous sections ξ k of E ( k ) . This convergence implies that ξ k +1 = ∇ ξ k inthe sense of distributions, hence by the equivalence of classical and distributionalderivatives (see e.g. [ LL01 , § η ∞ := ξ is smooth with ∇ k η ∞ = ξ k , so that ∇ k η j → ∇ k η ∞ in C ( E ( k ) ) for all k .We claim η ∞ ∈ C ε ( E ). Choose N > k η i − η j k C ε < i, j ≥ N .Then for every m ∈ N and every i ≥ N , m X k =0 ε k k η i k C k ≤ m X k =0 ε k k η i − η N k C k + m X k =0 ε k k η N k C k ≤ k η i − η N k C ε + k η N k C ε < k η N k C ε . Fixing m and letting i → ∞ , we then have m X k =0 ε k k η ∞ k C k ≤ k η N k C ε for all m , so we can now let m → ∞ and conclude k η ∞ k C ε ≤ k η N k C ε < ∞ .The argument that k η j → η ∞ k C ε → j → ∞ is similar: pick ǫ > N such that k η i − η j k C ε < ǫ for all i, j ≥ N . Then for a fixed m ∈ N , we can let i → ∞ in the expression P mk =0 ε k k η i − η j k C k < ǫ , giving m X k =0 ε k k η ∞ − η j k C k ≤ ǫ. This is true for every m , so we can take m → ∞ and conclude k η ∞ − η j k C ε ≤ ǫ forall j ≥ N . (cid:3) ectures on Symplectic Field Theory To show that C ε ( E ) is also separable, we will follow a hint from [ HS95 ] andembed it isometrically into another Banach space that can be more easily shown tobe separable. For each integer k ≥
0, define the vector bundle F ( k ) = E (0) ⊕ . . . ⊕ E ( k ) , and let X ε denote the vector space of all sequences ξ := ( ξ , ξ , ξ , . . . ) ∈ ∞ Y k =0 C ( F ( k ) )such that k ξ k X ε := ∞ X k =0 ε k k ξ k k C < ∞ . Exercise
B.3 . Adapt the proof of Theorem B.2 to show that X ε is also a Banachspace. Lemma
B.4 . X ε is separable. Proof.
Since C ( F ( k ) ) is separable for each k ≥
0, we can fix countable densesubsets P k ⊂ C ( F ( k ) ). The set P := (cid:8) ( ξ , . . . , ξ N , , , . . . ) ∈ X ε (cid:12)(cid:12) N ≥ ξ k ∈ P k for all k = 0 , . . . , N (cid:9) is then countable and dense in X ε . (cid:3) Theorem
B.5 . C ε ( E ) is separable. Proof.
Consider the injective linear map C ε ( E ) ֒ → X ε : η (cid:0) η, ( η, ∇ η ) , ( η, ∇ η, ∇ η ) , . . . (cid:1) . This is an isometric embedding and thus presents C ε ( E ) as a closed linear subspaceof X ε , hence the theorem follows from Lemma B.4 and the fact that subspaces ofseparable metric spaces are always separable. (cid:3) Note that given any open subset
U ⊂ M , Theorems B.2 and B.5 also hold for C ε ( E ; U ), as a closed subspace of C ε ( E ). So far in this discussion, however, there hasbeen no guarantee that C ε ( E ) or C ε ( E ; U ) contains anything other than the zero-section, though it is clear that in theory, one should always be able to enlarge thespace by choosing new sequences ε k that converge to zero faster. The following resultsays that C ε ( E ; U ) can always be made large enough to be useful in applications. Theorem
B.6 . Given an open subset
U ⊂ M , the sequence ε k can be chosen tohave the following properties:(1) C ε ( E ; U ) is dense in the space of continuous sections vanishing outside U .(2) Given any point p ∈ U , a neighborhood N p ⊂ U of p , a number δ > anda continuous section η of E , there exists a section η ∈ Γ( E ) and a smoothcompactly supported function β : N p → [0 , such that βη ∈ C ε ( E ; U ) , β ( p ) η ( p ) = η ( p ) , and k η − η k C < δ. Thanks to Sam Lisi for explaining to me what the hint in [
HS95 ] was referring to. Chris Wendl
Proof.
Note first that it suffices to find two separate sequences ε k and ε ′ k thathave the first and second property respectively, as the sequence of minima min( ε k , ε ′ k )will then have both properties.The following construction for the first property is based on a suggestion byBarney Bramham. Observe first that the space C ( E ; U ) of continuous sectionsvanishing outside U is a closed subspace of C ( E ) and is thus separable, so we canchoose a countable C -dense subset P ⊂ C ( E ; U ). Moreover, the space of smooth sections vanishing outside U is dense in C ( E ; U ), hence we can assume without lossof generality that the sections in P are smooth. Now write P = { η , η , η , . . . } anddefine ε k > k ≥ ε k < k min (cid:26) k η k C k , . . . , k η k k C k (cid:27) . Then every η j is in C ε ( E ; U ), as k η j k C ε < j − X k =0 ε k k η j k C k + ∞ X k = j k < ∞ . The second property is essentially local, so it can be deduced from Lemma B.7below. (cid:3)
Lemma
B.7 . Suppose β : ˚ D n → [0 , is a smooth function with compact supporton the open unit ball ˚ D n ⊂ R n and β (0) = 1 . One can choose a sequence of positivenumbers ε k → such that for every η ∈ R m and r > , the function η : R n → R m defined by η ( p ) := β ( p/r ) η satisfies P ∞ k =0 ε k k η k C k < ∞ . Proof.
Define ε k > k ≥ ε k = 1 k k k β k C k . Then ∞ X k =1 ε k k η k C k ≤ ∞ X k =1 k k k β k C k k β k C k r k = ∞ X k =1 (cid:18) /rk (cid:19) k < ∞ . (cid:3) PPENDIX C
Genericity in the space of asymptotic operators
The purpose of this appendix is to prove Lemma 3.17, which was needed forour definition of spectral flow in § H = L ( S , R n ) , D = H ( S , R n ) , the symmetric index 0 Fredholm operator T ref = − J ∂ t : D → H and, given a smooth loop of symmetric matrices S : S → End sym R ( R n ), refer to anyoperator of the form A = − J ∂ t − S : D → H as an asymptotic operator . Such operators belong to the space of symmetriccompact perturbations of T ref ,Fred sym R ( D , H , T ref ) = (cid:8) T ref + K : D → H (cid:12)(cid:12) K ∈ L sym R ( H ) (cid:9) , which we regard as a smooth Banach manifold via its obvious identification withthe space L sym R ( H ) of symmetric bounded linear operators on H . For k ∈ N , wedenote by Fred sym ,k R ( D , H , T ref ) ⊂ Fred sym R ( D , H , T ref )the finite-codimensional submanifold determined by the condition dim R ker A =dim R coker A = k .Here is the statement of Lemma 3.17 again. Lemma.
Fix a smooth map S : [ − , × S → End sym R ( R n ) and consider the -parameter family of operators A s := − J ∂ t − S ( s, · ) ∈ Fred sym R ( D , H , T ref ) for s ∈ [ − , . Then after a C ∞ -small perturbation of S fixed at s = ± , one canassume the following:(1) For every s ∈ ( − , , all eigenvalues of A s (regarded as an unboundedoperator on H ) are simple. Chris Wendl (2) All intersections of the path ( − , → Fred sym R ( D , H , T ref ) : s A s with Fred sym , R ( D , H , T ref ) are transverse. We shall now prove this by constructing a Floer-type space of C ε -smooth (seeAppendix B) perturbed families of asymptotic operators, and using the Sard-Smaletheorem to find a countable collection of comeager subsets whose intersection con-tains perturbations achieving the desired conditions.Choose a sequence of positive numbers ( ε ) ∞ k =0 with ε k → A ε := (cid:8) B ∈ C ∞ ([ − , × S , End sym R ( R n )) (cid:12)(cid:12) k B k C ε < ∞ and B ( ± , · ) ≡ (cid:9) , and assume via Theorem B.6 that A ε is dense in the Banach space of continuousfunctions [ − , × S → End sym R ( R n ) vanishing at {± } × S . We then considerperturbed 1-parameter families of asymptotic operators of the form A Bs := A s + B ( s, · ) : D → H for B ∈ A ε , s ∈ [ − , k ∈ N and B ∈ A ε , define the set V k ( B ) = (cid:8) ( s, λ ) ∈ ( − , × R (cid:12)(cid:12) dim R ker (cid:0) A Bs − λ (cid:1) = k (cid:9) . To show that eigenvalues are generically simple, we need to show that for a comeagerset of choices of B ∈ A ε , V k ( B ) is empty for all k ≥
2. Given ( s , λ ) ∈ V k ( B ),recall from § D = V ⊕ K, H = W ⊕ K where K = ker (cid:0) A Bs − λ (cid:1) , W = im (cid:0) A Bs − λ (cid:1) is the L -orthogonal complementof K , and V = W ∩ D , so that any symmetric bounded linear operator T in asufficiently small neighborhood O ⊂ L sym R ( D , H ) of A Bs − λ can be written inblock form T = (cid:18) A BC D (cid:19) with A : V → W invertible, giving rise to a smooth mapΦ : O →
End sym R ( K ) : T D − CA − B whose zero-set is precisely the set of nearby symmetric operators with k -dimensionalkernel. A neighborhood of ( s , λ ) in V k ( B ) can thus be identified with the zero-setof the map Ψ B ( s, λ ) := Φ( A Bs − λ ) ∈ End sym R ( K ) , defined for ( s, λ ) ∈ ( − , × R sufficiently close to ( s , λ ). Notice that the derivative d Ψ B ( s, λ ) : R ⊕ R → End sym R ( K ) is Fredholm since its domain and target are bothfinite dimensional, and it can only ever be surjective when k = dim R K = 1.The following space will now play the role of a “universal moduli space” as inLecture 7: let V k = (cid:8) ( s, λ, B ) ∈ ( − , × R × A ε (cid:12)(cid:12) ( s, λ ) ∈ V k ( B ) (cid:9) . ectures on Symplectic Field Theory The proof that this is a smooth Banach manifold depends on the following algebraiclemma.
Lemma
C.1 . Fix an asymptotic operator A = − J ∂ t − S and a linear transfor-mation Υ : ker A → ker A that is symmetric with respect to the L -product. Then there exists a smooth loop B : S → End sym ( R n ) such that h η, Bξ i L = h η, Υ ξ i L for all η, ξ ∈ ker A . Proof.
Note first that every nontrivial loop η ∈ ker A ⊂ H ( S , R n ) is smoothand nowhere zero since it satisfies a linear first-order ODE with smooth coeffi-cients. It follows that if we fix a basis ( η , . . . , η k ) for ker A , then the vectors η ( t ) , . . . , η k ( t ) ∈ R n are also linearly independent for all t ∈ S and thus spana smooth S -family of k -dimensional subspaces V t ⊂ R n , each equipped with adistinguished basis. It follows that there exists a unique smooth S -family of lineartransformations b B ( t ) : V t → V t such that for every η ∈ ker A , b B ( t ) η ( t ) = (Υ η )( t )for all t . Extend b B ( t ) arbitrarily to a smooth family of linear maps on R n .The matrices b B ( t ) ∈ End R ( R n ) need not be symmetric, but they do satisfy h η, b Bξ i L = h η, Υ ξ i L for all η, ξ ∈ ker A . Since Υ is symmetric, this implies moreover that for all η, ξ ∈ ker A , h η, Υ ξ i L = h ξ, Υ η i L = h ξ, b Bη i L = h η, b B T ξ i L . The loop B := ( b B + b B T ) thus has the desired properties. (cid:3) Now using the previously described construction in the space of symmetric Fred-holm operators, a neighborhood of any point ( s , λ , B ) in V k can be identified withthe zero-set of a smooth map of the formΨ( s, λ, B ) := Ψ B ( s, λ ) ∈ End sym F ( K ) , defined for all ( s, λ, B ) sufficiently close to ( s , λ , B ) in ( − , × R × A ε , where K = ker (cid:0) A B s − λ (cid:1) . The partial derivative of Ψ with respect to the third variableat ( s , λ , B ) is then a linear map L := D Ψ( s , λ , B ) : A ε → End sym R ( K )of the form(C.1) L B : K → K : η π K ( B ( s , · ) η ) , where π K : W ⊕ K → K is the orthogonal projection. We claim that L is surjective.Indeed, for any Υ ∈ End sym R ( K ), Lemma C.1 provides a smooth loop C : S → End sym R ( K ) such that π K ( C η ) = Υ η for all η ∈ K, and this can be extended to a smooth function C : [ − , × S → End sym R ( K )satisfying C ( s , · ) ≡ C and C ( ± , · ) ≡ s = ±
1. The function C might fail Chris Wendl to be of class C ε , but since it can be approximated arbitrarily well in the C -normby functions in A ε , we conclude that the image of L is dense in End sym R ( K ). Sincethe latter is finite dimensional, the claim follows.The implicit function theorem now gives V k the structure of a smooth Banachsubmanifold of ( − , × R × A ε , and it is separable since the latter is also separable.Consider the projection(C.2) π : V k → A ε : ( s, λ, B ) B, which is a smooth map of separable Banach manifolds whose fibers π − ( B ) are thespaces V k ( B ). Using Lemma 7.18, the fact that each map Ψ B is Fredholm impliesthat π is also a Fredholm map, so the Sard-Smale theorem implies that the regularvalues of π form a comeager subset A reg ,kε ⊂ A ε . The intersection A reg ε := \ k ∈ N A reg ,kε is then another comeager subset of A ε , with the property that for each B ∈ A reg ε and every ( s, λ ) ∈ V k ( B ), d Ψ B ( s, λ ) is (by Lemma 7.18) surjective. As was observedpreviously, this is impossible for dimensional reasons if k ≥
2, implying that V k ( B )is then empty.To find perturbations that also achieve the transversality condition, we use asimilar argument: define for each B ∈ A ε the subset V ( B ) = (cid:8) s ∈ ( − , (cid:12)(cid:12) dim R ker A Bs = 1 (cid:9) , along with the corresponding universal set V = (cid:8) ( s, B ) ∈ ( − , × A ε (cid:12)(cid:12) s ∈ V ( B ) (cid:9) . A neighborhood of any ( s , B ) in V is then the zero-set of a smooth map of theform Ψ( s, B ) = Φ( A Bs ) ∈ End sym R (ker A B s ) , defined for all ( s, B ) ∈ ( − , × A ε close enough to ( s , B ). For a fixed B ∈ A ε near B and s ∈ V ( B ) near s , a neighborhood of s in V ( B ) is then the zero-setof Ψ B ( s ) := Ψ( s, B ), and the intersection of the path s A s ∈ Fred sym R ( D , H , T ref )with Fred sym , R ( D , H , T ref ) at s = s is transverse if and only if d Ψ B ( s ) : R → End sym R (ker A B s )is surjective. At ( s , B ), the partial derivative of Ψ with respect to B is again thesame operator L = D Ψ( s , B ) : A ε → End sym R (ker A B s )as in (C.1), which we’ve already seen is surjective due to Lemma C.1. Thus one canapply the Sard-Smale theorem to the projection V → A ε : ( s, B ) B, obtaining a comeager subset A reg , ε ⊂ A ε such that all paths A s + B ( s, · ) for B ∈A reg , ε satisfy the required transversality condition. The comeager subset A reg , ε ∩ ectures on Symplectic Field Theory A reg ε ⊂ A ε thus consists of perturbed families of operators for which all desiredconditions are satisfied, and it contains a sequence converging in the C ∞ -topologyto 0. This concludes the proof of Lemma 3.17. ibliography [Abb14] C. Abbas, An introduction to compactness results in symplectic field theory , Springer,Heidelberg, 2014. MR3157146[AA02] Y. A. Abramovich and C. D. Aliprantis,
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