aa r X i v : . [ m a t h . S G ] F e b LOCAL SYMPLECTIC FIELD THEORY
OLIVER FABERT
Abstract.
Generalizing local Gromov-Witten theory, in this paper we definea local version of symplectic field theory. When the symplectic manifold withcylindrical ends is four-dimensional and the underlying simple curve is regu-lar by automatic transversality, we establish a transversality result for all itsmultiple covers and discuss the resulting algebraic structures.
Contents
Introduction 11. Local symplectic field theory 31.1. Symplectic field theory 31.2. A local version of symplectic field theory 51.3. Obstruction bundles from positivity of intersections 51.4. Transversality for multiple covers using obstruction bundles 91.5. Counting multiple covers of immersed curves with elliptic orbits 112. Application: Stable hypersurfaces intersecting exceptional spheres 142.1. Additional marked points and gravitational descendants 142.2. Obstruction bundle = normal bundle using topological recursion 152.3. Equations for the local SFT potentials 162.4. Exceptional spheres cannot break along hyperbolic orbits 18References 19
Introduction
In this paper we define a local version of Eliashberg-Givental-Hofer’s symplecticfield theory (SFT), see [EGH]. It provides a topological quantum theory approachto local Gromov-Witten theory in the same way as standard SFT provides atopological quantum field theory approach to standard Gromov-Witten theory.While in local Gromov-Witten theory one counts multiple covers over a fixedclosed holomorphic curve, see [LP], [BP], in local SFT we count multiple coversover punctured holomorphic curves.Instead of getting invariants for contact manifolds, we now get the invariantsfor closed Reeb orbits that were already studied in [F2] and [F3]. Note that forthe orbit curves we used an infinitesimal energy estimate to show that multiplecovers of orbit cylinders are isolated in the moduli space of holomorphic curves.In this paper we show that the dimension bounds on the kernel of the linearizedCauchy-Riemann operator established in [Wen] (using positivity of intersections indimension four) can be used to obtain the required isolatedness result for rationalmultiple covers when the underlying simple (rational) curve is sufficiently nice.
Theorem 0.1.
Assume that the rigid holomorphic curve v : S → X is immersedand that all asymptotic orbits are Morse nondegenerate and elliptic. If ind u = 0 for rational multiple covers u = v ◦ ϕ in M v,d (Γ + , Γ − ) , then every infinitesimaldeformation of u as a holomorphic curve is again a multiple cover of v . Further-more the cokernels of the linearized Cauchy-Riemann operator ¯ ∂ J fit together toa smooth obstruction bundle Coker v ¯ ∂ J = Coker v,d (Γ + , Γ − ) over the compactifiedmoduli space M v = M v,d (Γ + , Γ − ) . Using these obstruction bundles we can solve the transversality problem formultiply-covers of immersed curves with elliptic orbits without employing the poly-fold machinery from [HWZ], see ([MDSa], section 7.2) for the general approach.
Proposition 0.2.
Let ν be a section in the cokernel bundle Coker ¯ ∂ J ⊂ E | M overthe moduli space M = ¯ ∂ − J (0) ⊂ B , which is extended (using parallel transport andcut-off functions, as described in [F2] , [MDSa] , [LP] ) to a section in the full Banachspace bundle E → B . Then it holds: • The perturbed moduli space M ν = ( ¯ ∂ J + ν ) − (0) agrees with the zero set of ν in M , M ν = ν − (0) . • If ν is a transversal section in Coker ¯ ∂ J , then ¯ ∂ νJ is a transversal section in E , i.e., M ν is regular. • The linearization of ν at every zero is a compact operator, so that thelinearizations of ¯ ∂ J and ¯ ∂ νJ belong to the same class of Fredholm operators. We then show how this can be used to define morphisms in the local versionof Eliashberg-Givental-Hofer’s symplectic field theory introduced by the authorin [F2]. More precisely, we show that immersed holomorphic curves with ellip-tic asymptotic orbits in four-dimensional symplectic cobordisms define morphismsbetween the local SFT invariants assigned to their asymptotic closed Reeb orbits.While in standard SFT one collects the information about all moduli spaces of holo-morphic curves in X by defining a potential f , we now define a local SFT potential f v ∈ L Γ ′ + , Γ ′− counting only multiple covers of the fixed rigid immersed curve withelliptic orbits v : S → X . Definition 0.3.
For every choice of obstruction bundle sections (¯ ν ) coherentlyconnecting the coherent collections of obstruction bundle sections (¯ ν ± ) chosen forall positive and negative asymptotic Reeb orbits γ ± ∈ Γ ′± of v , we define the localSFT potential of a rigid immersed holomorphic curve v with elliptic orbits by f v ! = f (¯ ν ) v = X Γ + , Γ − s + ! s − ! κ Γ + κ Γ − M ¯ νv,d (Γ + , Γ − ) q Γ − − p Γ + + , where M ¯ νv,d (Γ + , Γ − ) = ¯ ν − (0) ⊂ M v,d (Γ + , Γ − ) . Furthermore we will use the result in [F2] to discuss how the algebraic count ofmultiple covers of a immersed curve with elliptic orbits depends on all the auxiliarychoices. Here we prove the following
Theorem 0.4.
Assume that the coherent collections of sections (¯ ν ± ) are fixed forall asymptotic Reeb orbits γ ± ∈ Γ ′± of v . Then the local SFT potential f v = f (¯ ν ) v of v is independent of the chosen collection of sections (¯ ν ) coherently connecting (¯ ν + ) and (¯ ν − ) . In particular, the algebraic count of multiple covers of the immersedcurve with elliptic orbits v is well-defined. At the end we illustrate how the new local SFT invariants can be used to obtainricher obstructions against stable embeddings of hypersurfaces in four-dimensional ocal SFT 3 symplectic manifolds. After introducing additional marked points and gravitationaldescendants (translated into branching conditions as in [F3]), we reprove the fol-lowing result due to Welschinger.
Theorem 0.5.
Assume that a closed oriented Lagrangian surface L in a closedsymplectic four-manifold has a homologically nontrivial intersection with an excep-tional sphere Σ . Then L must be diffeomorphic to S or S × S . This paper is organized as follows: After recalling the basic definitions andresults of symplectic field theory in subsection 1.1, we discuss the ideas and maindefinitions of its local version in subsection 1.2. While in subsection 1.3 we use theresults in [Wen] to prove the desired isolatedness result for multiple covers whichalso proves the existence of a smooth finite-dimensional obstruction bundle overthe corresponding moduli spaces, we show in 1.4 how the transversality problemfor the Cauchy-Riemann operator can be solved by choosing transversal andcoherent sections in these bundles and discuss in 1.5 how the resulting algebraiccount of multiple covers depends on these auxiliary choices. In section two wethen show how local SFT methods can be applied to stable embedding problems ofhypersurfaces in symplectic blow-ups. After introducing gravitational descendantsvia branching conditions in 2.1, we explicitly compute a contribution to the localGromov-Witten descendant potential of an exceptional sphere using topologicalrecursion in 2.2. In subsection 2.3 we then use our computation to prove equationsfor the local SFT potential and finally prove our obstruction to stable embeddingsof hypersurfaces in 2.4.The author is deeply indebted to Chris Wendl for explaining to him his workon automatic transversality in long discussions, which were the starting point forthis project. Furthermore he wants to thank Kai Cieliebak for his ideas concerningthe generalization of the result in [F3] and also thanks E. Ionel, T. Parker andC. Taubes for interesting discussions on this topic during his stay at the MSRIin Berkeley. This paper was written when the author was a research assistantat the University of Augsburg. He is grateful for the financial support and thegreat working environment and also thanks Prof. K. Wendland’s ERC StartingIndependent Researcher Grant (StG No. 204757-TQFT) for further support.1.
Local symplectic field theory
Symplectic field theory.
Symplectic field theory was defined by Eliashberg,Givental and Hofer in their paper [EGH] and is designed to describe in a unifiedway the theory of pseudoholomorphic curves in symplectic and contact topology.In particular, it defines a functor from a geometric category to an algebraiccategory. The objects of the geometric category are contact manifolds (moregenerally, manifolds with stable Hamiltonian structure) V of dimension 2 n − n ≥
1) , while the morphisms from one contact manifold V − to another contactmanifold V + are strong symplectic cobordisms X of dimension 2 n from V − to V + , that is, strong symplectic fillings of the disconnected union − V − ∪ V + .The functor SFT defines invariants for contact manifolds V , denoted by SFT( V ),by counting J -holomorphic curves in cylindrical manifolds R × V equipped with acompatible almost complex structure J , which is cylindrical in the sense that itis R -invariant, preserves the contact distribution, ξ = ker λ = T V ∩ JT V , andmaps the R -direction to the Reeb vector field R ∈ ker dλ , λ ( R ) = 1. For thisrecall that a contact one-form λ defines a vector field R on V by R ∈ ker dλ and λ ( R ) = 1, which is called the Reeb vector field. Throughout the paper we assume
O. Fabert that the contact form is Morse in the sense that all closed orbits of the Reeb vectorfield are (Morse) nondegenerate in the sense that one is not an eigenvalue of thelinearized return map; in particular, the set of closed Reeb orbits is discrete.
Theinvariants are defined by counting J -holomorphic curves u in R × V . Let Γ + , Γ − betwo ordered sets of closed (unparametrized) orbits γ of the Reeb vector field R on V .Note further that in this paper we just restrict to the case of rational holomorphiccurves. Then the (parametrized) moduli space M V,A (Γ + , Γ − ) consists of tuples( u, j ), where j is a complex structure on the sphere S = S − { z ± , ..., z ± s ± } with s = s + + s − punctures ( s ± = ± ) removed and maps u : ( S, j ) → ( R × V, J )satisfying the Cauchy-Riemann equation¯ ∂ J u = du + J ( u ) · du · j = 0 . Assuming we have chosen cylindrical holomorphic coordinates ψ ± k : R ± × S → ( S, j ) around each puncture z ± k in the sense that ψ ± k ( ±∞ , t ) = z ± k , the map u isadditionally required to show for all k = 1 , ..., n ± the asymptotic behaviourlim s →±∞ ( u ◦ ψ ± k )( s, t + t ) = ( ±∞ , γ ± k ( T ± k t ))with some t ∈ S and the orbits γ ± k ∈ Γ ± , where T ± k > γ ± k . In particular, note that in the asymptotic condition is independent of theparametrization of the closed Reeb orbit. In order to assign an absolute homologyclass A ∈ H ( V ) to a holomorphic curve u : ( S, j ) → ( R × V, J ) we have to employspanning surfaces u γ connecting a given closed Reeb orbit γ in V to a linearcombination of circles c s representing a basis of H ( V ), ∂u γ = γ − P s n s · c s inorder to define A = [ u Γ + ] + [ u ( S )] − [ u Γ − ], where [ u Γ ± ] = P s ± n =1 [ u γ ± n ] viewed assingular chains.Observe that when the number of punctures is less than three the correspondingsubgroup Aut( S, j ) with (
S, j ) = R × S , C , CP of the group of Moebius transfor-mations acts on elements in M V,A (Γ + , Γ − ) in an obvious way, ϕ. ( u, j ) = ( u ◦ ϕ − , j ) , ϕ ∈ Aut(
S, j ) , and we obtain the moduli space M = M V,A (Γ + , Γ − ) studied in symplectic fieldtheory by dividing out this action and the natural R -action on the target manifold( R × V, J ).To every strong symplectic cobordism X from V − to V + , the functor SFT as-signes morphisms SFT( X ) from the invariant SFT( V − ) to the invariant SFT( V + )by counting J -holomorphic curves in X , where the ω -compatible almost complexstructure J agrees with J ± on the cylindrical ends R ± × V ± of X . For the latterobserve that here and in what follows we do not distinguish between the strongsymplectic filling and its (non-compact) completion. Indeed, let X = ( X, ω ) be asymplectic manifold with cylindrical ends ( R + × V + , λ + ) and ( R − × V − , λ − ) in thesense of ([BEHWZ], section 3) which is equipped with an almost complex structure J which agrees with the cylindrical almost complex structures J ± on R + × V + .Then we study J -holomorphic curves u : ( S, j ) → ( X, J ) which are asymptoticallycylindrical over chosen collections of orbits Γ ± = { γ ± , ..., γ ± n ± } of the Reeb vectorfields R ± in V ± as the R ± -factor tends to ±∞ , see [BEHWZ]. We now denoteby M X,A (Γ + , Γ − ), M X,A (Γ + , Γ − ) and M X,A (Γ + , Γ − ) the corresponding modulispaces of rational curves in X , where it is important to note that for passing from M X,A (Γ + , Γ − ) to M X,A (Γ + , Γ − ) we do no longer divide out a symmetry on targetanymore. After choosing abstract perturbations using polyfolds as described above,we again find that M X,A (Γ + , Γ − ) is a weighted branched manifold with boundaries ocal SFT 5 and corners of dimension equal to the Fredholm index of the Cauchy-Riemannoperator for J .Furthermore it was shown in ([BEHWZ], theorem 10.1 and 10.2) that M V,A (Γ + , Γ − ) and M X,A (Γ + , Γ − ) can be compactified to moduli spaces M V,A = M V,A (Γ + , Γ − ) and M X,A = M X,A (Γ + , Γ − ) by adding moduli spaceof multi-floor curves with nodes. After choosing abstract perturbations usingpolyfolds (see [HWZ]) we get that M V,A and M X,A is a branched-labelled manifoldwith boundaries and corners of dimension equal to the Fredholm index of theCauchy-Riemann operator for J (minus one in the first case, where the latteraccounts for dividing out the one-dimensional R -action on the target). In partic-ular, the moduli spaces M V,A and M X,A have codimension-one boundary givenby (fibre) products M V,A × M V,A and M X,A × M V − ,A ∪ M V + ,A × M X,A ( A + A = A ) of lower-dimensional moduli spaces, respectively.1.2. A local version of symplectic field theory.
Note that for n = 1 theone-dimensional contact manifold V consists of a copies of circles, while a two-dimensional symplectic cobordism from V − to V + is nothing else but a Riemannsurface S with s − negative and s + positive punctures, i.e., points removed, where s ± denotes the number of components of V ± . While the SFT-invariants for V = S count branched coverings of the cylinder R × S , the morphism SFT( S ) isdefined by counting branched coverings of S .While for n = 1 the SFT functor is easily understood, researchers were lookingfor computable examples, which can be viewed as an intermediate step betweenthe case of Riemann surfaces and the case of general symplectic manifolds. Andindeed, it is well-known in Gromov-Witten theory that one can define a local version of it, see [BP] and [LP], by counting multiple covers of a fixed (simple)rigid J -holomorphic curve v : ( S , i ) → ( X, J ). Indeed it can be shown that undercertain assumptions on v the submoduli spaces M v,d of d -fold branched coverings u = v ◦ ϕ : ( S , i ) → ( S , i ) → ( X, J ) is a connected component of the modulispace M X,d [ v ] of general holomorphic maps to X .In this paper we define a local version of symplectic field theory, which providesa TQFT approach to local Gromov-Witten theory in the same way as standardsymplectic field theory provides a TQFT approach to standard Gromov-Wittentheory.The corresponding invariants SFT( γ ) for closed Reeb orbits γ were introducedby the author in [F2], [F3] by counting holomorphic curves in the moduli spaces M γ,d (Γ + , Γ − ) ⊂ M V, (Γ + , Γ − ) of branched covers ϕ : ( S, j ) → ( R × γ, J ). In thepresent paper we show that immersed holomorphic curves with elliptic orbits v canbe used to define morphisms SFT( v ) between the invariants SFT(Γ + ) and SFT(Γ − )assigned to its asymptotic orbits. For this we again show that the submoduli spaces M v,d (Γ + , Γ − ) ⊂ M X,d [ v ] (Γ + , Γ − )of multiple covers u = v ◦ ϕ : ( S, j ) → ( S ′ , j ′ ) → ( X, J ) are isolated in the space ofall holomorphic maps in the sense that every infinitesimal deformation of a multiplecover in M v,d (Γ + , Γ − ) as a holomorphic curve is still a multiple cover.1.3. Obstruction bundles from positivity of intersections.
We denote by M v = M v,d (Γ + , Γ − ) the moduli space of parametrized branched coverings u = O. Fabert v ◦ ϕ : ( S, j ) → ( S ′ , j ′ ) → ( X, J ) of a fixed holomorphic map v : ( S ′ , j ′ ) → ( X, J ),.Note that for here we do not yet divide out any symmetry of the domain. As in[F2], for establishing the desired isolatedness result we need to prove that T u M v = ker D u , see also ([MDSa], section 7.2). Here D u : T u B → E u denotes the linearization ofthe Cauchy-Riemann operator ¯ ∂ J , viewed as a smooth section in an appropriateBanach space bundle E → B with fibre E u = L p,d (Λ , ⊗ j,J u ∗ T X ) over the Banachmanifold of maps B with tangent space T u B = H ,p,d ( u ∗ T X ) ⊕ T j M ,n , see [BM],where the second summand keeps track of the variation of the complex structureon S .Note that we always have the inclusion T u M v ⊂ ker D u . Assume that theunderlying holomorphic curve v : ( S ′ , j ′ ) → ( X, J ) is simple and J is a genericalmost complex structure on X . By the well-known transversality result for simpleholomorphic curves, it follows that the local dimension of M v near u = v ◦ ϕ isgiven by dim T u M v = ind( v ) + 2 ϕ ) , where ϕ ) denotes the number of branch points of the branched covering map ϕ : ( S, j ) → ( S ′ , j ′ ). Note that the latter number is fully determined by Γ + and Γ − .Since ind u ≤ dim ker D u , it follows that the desired equality T u M v = ker D u can only hold when ind u ≤ ind v + 2 ϕ ). While the indexind u = dim ker D u − dim coker D u can be computed from topological data,in particular, is constant over each connected component of the moduli space ofholomorphic curves, the dimensions of kernel and cokernel themselves usuallyjump and are very hard to be controlled in general.It was shown in [Wen] that in dim X = 4 the dimensions of ker D u and coker D u can be controlled by topological data making use of positivity of intersections. Thiswas used to prove an automatic transversality result for so-called nicely-embeddedholomorphic curves by showing that the topological bounds imply coker D u = 0for any choice of almost complex structure J . We will now show that the sameinequalities can be used to prove the following improvement about multiple covers.Recall that in a three-dimensional contact manifold all closed Reeb orbits areeither elliptic or hyperbolic. Here an orbit is called hyperbolic if all eigenvalues ofthe linearized return map are real and elliptic if they lie on the unit circle in thecomplex plane. Furthermore a (simple) holomorphic curve v is called rigid if theFredholm index of v is zero. Theorem 1.1.
Assume that the rigid holomorphic curve v : S → X is immersedand that all its asymptotic orbits are elliptic. Then if ind u ≤ dim T u M v =2 ϕ ) we have T u M v = ker D u , i.e., every infinitesimal deformation of u as a holomorphic curve is again a multiple cover of v . Before we give the proof using the results from [Wen], we remark that this resultindeed contains the automatic transversality result from [Wen] as a special case (forindex zero curves and up to excluding odd hyperbolic orbits because of troubleswith bad orbits). Indeed it follows directly from the definition of nicely-embeddedcurves in ([Wen], definition 4.12) that each such holomorphic curve is immersedand has only odd orbits, see ([Wen], section 4.3). When u = v , i.e., ϕ is the trivialcovering, then ker D u = 0(= ind( u )) implies that u is regular. ocal SFT 7 Proof.
As mentioned above, we show the desired result using the following results inthe paper [Wen]. First, in ([Wen], section 3.3) it is shown that for every holomorphiccurve u : ( S, j ) → ( X, J ) there exists a splitting of the pull-back bundle u ∗ T X = T u ⊕ N u , where T u and N u denote the tangent and the normal bundle to u , respectively.While T u agrees with du ( T S ) away from the critical points of u , the normal bundle N u agrees with the contact hyperplane distributions ξ ± = T V ± ∩ J ± T V ± in thecylindrical ends ( R ± × V ± , J ± ) of ( X, J ).Using this splitting, in ([Wen], section 3.4) the author introduces the normalCauchy-Riemann operator D Nu : H ,p,d ( N u ) ⊕ T j M ,n → L p,d (Λ , ⊗ j,J N u ) . Furthermore he shows in ([Wen], theorem 3) that the dimensions of kernel andcokernel of D u and D Nu are related bydim ker D u = 2 u ) + dim ker D Nu and coker D u = coker D Nu , where Crit( u ) denotes the set of critical points of u .Since the underlying simple curve v is immersed, we have u ) = ϕ ) = dim T u M v . Hence it remains to prove that ker D Nu = 0, which can be shown using the estimatesin [Wen].First, following ([Wen], section 1.1) the normal first Chern number of u is definedby 2 c N ( u ) = ind u − g + = ind u − , where = 0 denotes the number of even asymptotic orbits of u and g denotesthe genus of u . Note that the number of even asymptotic orbits of u is indeed zeroas every multiply-covered elliptic orbit is still elliptic and hence odd.Following ([Wen], proposition 3.18) the normal first Chern number is related toan adjusted first Chern number c ( N u ) of the normal bundle by c ( N u ) = c N ( u ) − u ) , where again Crit( u ) = Crit( ϕ ) agrees with the set of branch points. Here ”ad-justed” refers to the fact that the count of zeros involves asymptotic intersections.Since ind( u ) ≤ dim T u M v = 2 ϕ ), note that as in the proof of ([Wen],theorem 1) we still have c ( N u ) <
0, independent of the number of branch points.While the definition of c ( N u ) is quite complicated, all we need for our proof is thatit is shown in ([Wen], proposition 2.2 (1)) that the latter implies ker D Nu = 0. (cid:3) In particular, the latter holds true for all multiple covers of immersed holomor-phic curves v with elliptic orbits which are virtually rigid, i.e., with ind u = 0. Asin ([MDSa], section 7.2) we can use our bound on the dimension of ker D u to provethe existence of an obstruction bundle of constant rank. Corollary 1.2. If ind u = 0 for u ∈ M v,d (Γ + , Γ − ) , then the cokernels of thelinearized Cauchy-Riemann operator ¯ ∂ J fit together to a smooth obstruction bundleCoker v ¯ ∂ J = Coker v,d ¯ ∂ J (Γ + , Γ − ) over the moduli space of multiple covers M v = M v,d (Γ + , Γ − ) . O. Fabert
Proof.
From the desired equality T u M v = ker D u it follows that the cokernelcoker D u has dimensiondim coker D u = dim ker D u − ind u = ind v + 2 ϕ ) − ind u, in particular, is constant over each connected component of the moduli space ofmultiple covers. (cid:3) In [F2] the author has shown that such an obstruction bundle exists over themoduli space of multiple covers over each orbit cylinder R × γ in R × V . Theneccessary equality T u M γ = ker D u for all moduli spaces M γ = M γ,d (Γ + , Γ − ) oforbit curves was proven in [F2] using energy considerations, since it is the infini-tesimal version of the statement in [BEHWZ] that every holomorphic curve withzero contact area (in the sense of [BEHWZ]) is a multiple cover of an orbit cylinder.Further it follows from the compactness results from [BEHWZ] stated above(see also [F2]) that the codimension-one boundary of the compactified modulispace M γ,d (Γ + , Γ − ) of multiple covers of the orbit cylinder R × γ is given by(fibre) products M γ,d (Γ +1 , Γ − ) × M γ,d (Γ +2 , Γ − ) of lower-dimensional modulispaces of multiple covers over the same orbit cylinder. Note that here we workwith the unperturbed moduli space, which agrees with the moduli space for thecontact manifold S , so everything reduces to studying branched coverings ofcylinders and all curves are automatically regular (as a map to the cylinder over S ).The same compactness result proves that the codimension-one bound-ary of the compactified moduli space M v,d (Γ + , Γ − ) of multiple covers overa rigid immersed curve with elliptic orbits v is given by (fibre) products M γ + ,d + (Γ +1 , Γ − ) × M v,d (Γ +2 , Γ − ) and M v,d (Γ +1 , Γ − ) × M γ − ,d − (Γ +2 , Γ − ) of lower-dimensional moduli spaces, where on the zero level we still consists of multiplecovers over the original rigid immersed curve with elliptic orbits, while on thepositive and negative levels we find multiple covers over cylinders over asymptoticReeb orbits γ + and γ − for v , respectively.In order to compute the new SFT invariants for closed Reeb orbits, it was furthershown in [F2] that the obstruction bundle actually extends to the compactifiedmoduli space, which again follows from energy reasons and a linear gluing theorem.We now prove that also the above obstruction bundle Coker v ¯ ∂ J = Coker v,d (Γ + , Γ − )over the moduli space of multiple covers of a rigid immersed curves with only ellipticasymptotic orbits extends to the compactification. Theorem 1.3.
The obstruction bundle
Coker v ¯ ∂ J = Coker v,d (Γ + , Γ − ) over themoduli space M v = M v,d (Γ + , Γ − ) of index zero multiple covers of a rigid immersedcurve with elliptic orbits extends to a smooth bundle Coker v ¯ ∂ J = Coker v,d (Γ + , Γ − ) over its compactification M v = M v,d (Γ + , Γ − ) .Proof. Assume that the sequence of multiple covers u n = v ◦ ϕ n ∈ M v,d (Γ + , Γ − )converges to a two-level curve ( u + , u ) ∈ M γ + ,d + (Γ +1 , Γ − ) × M v,d (Γ +2 , Γ − ) in thesense of [BEHWZ]. Following [F2] it can be shown using work of Long aboutthe Conley-Zehnder index of multiply-covered Reeb orbits that ind u + ≥ X = 4. Together with ind u + + ind u = ind u = 0 we get ind u ≤ u still meets the requirements from the above theorem and we have T u M v = ker D u . On the other hand, from [F2] we get by energy considera-tions that T u + M γ + = ker D u + . Putting together, we find that also for the brokencurve ( u + , u ) ∈ M γ + × M v in the compactification we have T ( u + ,u ) ( M γ + × M v ) = T u + M γ + ⊕ T u M v = ker D u + ⊕ ker D u = ker D ( u + ,u )ocal SFT 9 as desired, so that we can prove the existence of the smooth bundle Coker v ¯ ∂ J usinglinear gluing as in [F2]. (cid:3) Transversality for multiple covers using obstruction bundles.
Aftershowing that for the moduli spaces of multiple covers of rigid immersed curveswith elliptic orbits we have the same nice obstruction bundle setup as for themoduli spaces of multiple covers over orbit cylinders, we now want to discuss theappearing algebraic structures. In other words, we want to discuss in how far onecan actually count multiple covers of these nice curves, i.e., in how far this countdepends on chosen auxiliary data like abstract perturbations needed in order toachieve regularity.Before all that, we however need to state the main theorem about obstructionbundle transversality, see ([F2], proposition 3.1). For alternative proofs we refer to([MDSa], proposition 7.2.3) and the proof of the main theorem in [LP].
Proposition 1.4.
Let ν be a section in the cokernel bundle Coker ¯ ∂ J ⊂ E | M overthe moduli space M = ¯ ∂ − J (0) ⊂ B , which is extended (using parallel transport andcut-off functions, as described in [F2] , [MDSa] , [LP] ) to a section in the full Banachspace bundle E → B . Then it holds: • The perturbed moduli space M ν = ( ¯ ∂ J + ν ) − (0) agrees with the zero set of ν in M , M ν = ν − (0) . • If ν is a transversal section in Coker ¯ ∂ J , then ¯ ∂ νJ is a transversal section in E , i.e., M ν is regular. • The linearization of ν at every zero is a compact operator, so that thelinearizations of ¯ ∂ J and ¯ ∂ νJ belong to the same class of Fredholm operators. Note that the above theorem does not only hold for the non-compact modulispace itself, but also for the other moduli spaces of orbit curves appearing in thecompactification. Using the linear gluing result in [F2] it follows that a compacti-fication of the perturbed moduli space M ν is given by M ¯ ν = ¯ ν − (0) ⊂ M for a smooth section ¯ ν in the extended obstruction bundle Coker ¯ ∂ J over thecompactified nonregular moduli space M of multiple covers of the orbit cylinder.Note that in order to formulate the corresponding version of the above propositiondirectly for the compactified moduli space, one has to work with polyfolds insteadof Banach manifolds.In Gromov-Witten theory we would hence obtain the contribution of the regularperturbed moduli space by integrating the Euler class of the finite-dimensionalobstruction bundle over the compactified moduli space. On the other hand, passingfrom Gromov-Witten theory back to symplectic field theory again, we see that thepresence of codimension-one boundary of the nonregular moduli spaces of branchedcovers implies that Euler numbers for sections in the cokernel bundles are notdefined in general, since the count of zeroes depends on the compact perturbationschosen for the moduli spaces in the boundary. Instead of looking at a singlemoduli space, we have to consider all moduli spaces at once and define coherent collections of sections in the obstruction bundles Coker ¯ ∂ J over all moduli spaces M .We start with the case of multiple covers of orbit cylinders. Recall from [F2]that the codimension-one boundary of every moduli space M = M γ,d (Γ + , Γ − )again consists of curves with two levels, whose moduli spaces can be represented as products M × M = M γ,d (Γ +1 , Γ − ) × M γ,d (Γ +2 , Γ − ) of moduli spaces of strictlylower dimension, where the first index refers to the level. On the other hand, itfollows from the linear gluing result in [F2] that over the boundary component M × M the cokernel bundle Coker ¯ ∂ J = Coker γ,d ¯ ∂ J (Γ + , Γ − ) is given byCoker ¯ ∂ J | M ×M = π ∗ Coker ¯ ∂ J ⊕ π ∗ Coker ¯ ∂ J , where Coker , ¯ ∂ J = Coker γ,d , ¯ ∂ J (Γ +1 , , Γ − , ), denote the cokernel bundles over thecompact moduli spaces M , = M γ,d , (Γ +1 , , Γ − , ) and π , : M , M → M , isthe projection onto the first or second factor, respectively.Assume that we have chosen sections ¯ ν = ¯ ν γ,d (Γ + , Γ − ) in the cokernel bundlesCoker ¯ ∂ J over all moduli spaces M of branched covers. Following ([F2], definition3.2) we call this collection of sections (¯ ν ) coherent if over every codimension-oneboundary component M × M of a moduli space M = M γ,d (Γ + , Γ − ) thecorresponding section ¯ ν agrees with the pull-back π ∗ ¯ ν ⊕ π ∗ ¯ ν of the chosensections ¯ ν , in the cokernel bundles Coker , ¯ ∂ J over M , , respectively.Since in the end we will again be interested in the zero sets of these sections,we will again assume that all occuring sections are transversal to the zero section.On the other hand, it is not hard to see that one can always find such coherentcollections of (transversal) sections in the cokernel bundles by using induction onthe dimension of the underlying nonregular moduli space of branched covers.Now we want to turn to the moduli spaces of multiple covers of immersedcurves with elliptic orbits in four-dimensional symplectic cobordisms. Recallthat the codimension-one boundary of every moduli space of branched covers M = M v,d (Γ + , Γ − ) again consists of curves with two levels, whose moduli spacescan be represented as products M + × M = M γ + ,d + (Γ +1 , Γ − ) × M v,d (Γ +2 , Γ − )and M × M − = M v,d (Γ +1 , Γ − ) × M γ − ,d − (Γ +2 , Γ − ) of moduli spaces of strictlylower dimension. On the other hand, it follows that over the boundary component M + × M or M × M − the cokernel bundle Coker ¯ ∂ J = Coker v,d ¯ ∂ J (Γ + , Γ − ) isgiven by Coker ¯ ∂ J | M + ×M = π ∗ + Coker + ¯ ∂ J ⊕ π ∗ Coker ¯ ∂ J , Coker ¯ ∂ J | M ×M − = π ∗ Coker ¯ ∂ J ⊕ π ∗− Coker − ¯ ∂ J , where Coker ¯ ∂ J = Coker v,d ¯ ∂ J (Γ +1 , , Γ − , ) and Coker + ¯ ∂ J = Coker γ + ,d + ¯ ∂ J (Γ +1 , Γ − ),Coker − ¯ ∂ J = Coker γ − ,d − ¯ ∂ J (Γ +2 , Γ − ) denote the cokernel bundle over the modulispace M of multiple covers of the immersed curves with elliptic orbits and themoduli spaces M + , M − of multiple covers of cylinders over positive or negativeasymptotic Reeb orbits γ ± of v , respectively.With this we can now give the analogue of the above definition of special sections¯ ν = ¯ ν v,d (Γ + , Γ − ) in obstruction bundles Coker ¯ ∂ J = Coker v,d ¯ ∂ J (Γ + , Γ − ) over mod-uli spaces M = M v,d (Γ + , Γ − ) of multiple covers of immersed curves with elliptic or-bits v . Assume that we have already coherently chosen sections ¯ ν ± = ¯ ν γ ± ,d (Γ + , Γ − )in the cokernel bundles Coker ± ¯ ∂ J = Coker γ ± ,d ¯ ∂ J (Γ + , Γ − ) over all moduli spaces M ± = M γ,d (Γ + , Γ − ) of branched covers of cylinders over positive and negativeasymptotic Reeb orbits γ ± of v . ocal SFT 11 Definition 1.5.
Assume that we have chosen sections ¯ ν in the cokernel bundles Coker ¯ ∂ J over all moduli spaces M of multiple covers of the immersed curve withelliptic orbits v . Then we call such a collection of sections (¯ ν ) coherently connecting (¯ ν + ) and (¯ ν − ) if over every codimension-one boundary component M + ×M , M ×M − the corresponding section ¯ ν agrees with the pull-back π ∗ + ¯ ν + ⊕ π ∗ ¯ ν , π ∗ ¯ ν ⊕ π ∗− ¯ ν − of the chosen sections ¯ ν and ¯ ν + , ¯ ν − in the cokernel bundles Coker ¯ ∂ J over M , Coker ± ¯ ∂ J over M ± , respectively. Since in the end we will again be interested in the zero sets of sections, we willagain assume that all occuring sections are transversal to the zero section. Onthe other hand, it is again not hard to see that one can always find such coherentcollections of (transversal) sections in the cokernel bundles by using induction onthe dimension of the underlying nonregular moduli space of branched covers.1.5.
Counting multiple covers of immersed curves with elliptic orbits.
Wenow turn to the resulting algebraic structures, where we first recall the algebraicformalism to define invariants for closed Reeb orbits. Denote by P γ be the gradedPoisson subalgebra of the Poisson algebra P of rational SFT, which is generatedonly by those p - and q -variables p γ n , q γ n corresponding to Reeb orbits which aremultiple covers of the fixed orbit γ and which are good in the sense of [BM]. It willbecome important that the natural identification of the formal variables p γ n and q γ n for different orbits γ does not lead to an isomorphism of the graded algebras P γ with the corresponding graded algebra P S for γ = V = S , not only sincethe gradings of p γ n and q γ n are different and hence even the commutation rulesmay change but also that variables p γ n and q γ n may not be there since they wouldcorrespond to bad orbits.In ([EGH], section 2.2.3) one collects the information about all moduli spacesof holomorphic curves in R × V in a generating function, the SFT Hamiltonian h , which does not only depend on contact form and cylindrical almost complexstructure but also on the collection of abstract perturbations. As in [F2] we nowdefine a local SFT Hamiltonian h γ ∈ P γ by only counting branched covers of thecylinder over the Reeb orbit γ . Instead of working with polyfold perturbations,we have seen above that we can make all moduli spaces of orbit curves regularby choosing sections in the cokernel bundles over all moduli spaces. For such acollection of sections (¯ ν ) we then define the Hamiltonian h γ = h (¯ ν ) γ by h (¯ ν ) γ = X Γ + , Γ − s + ! s − ! κ Γ + κ Γ − M ¯ νγ,d (Γ + , Γ − ) q Γ − p Γ + , with p Γ + = p γ n +1 . . . p γ n + s + and q Γ − = q γ n − . . . q γ n − s − , where M ¯ νγ,d (Γ + , Γ − ) = ¯ ν − (0) ⊂ M γ,d (Γ + , Γ − ) . Furthermore s ± = ± and κ Γ ± = κ γ n − . . . κ γ n − s − , where κ γ denotes the multi-plicity of the orbit γ .Note that in general we have to expect that the local SFT Hamiltonian explicitlydepends on the chosen coherent collection of sections. However, in [F2] we wereable to prove the following result. Theorem 1.6.
For every closed Reeb orbit γ the Hamiltonian h γ = h ¯ νγ vanishes in-dependently of the chosen coherent collection of sections (¯ ν ) in the cokernel bundlesover all moduli spaces of branched covers, h γ = h ¯ νγ = 0 . Although the result of our computation may suggest that it follows a globalsymmetry of the resulting regular moduli space, we want to emphasize that the S -action on the underlying nonregular moduli space of branched covers in generaldoes not lift to an action on the obstruction bundle over this space, so that theresulting perturbed moduli space does not carry a global symmetry.In the same way as for a single orbit we define for collections of Reeb orbits Γ thePoisson algebras P Γ to be the graded Poisson subalgebras of the Poisson algebra P , which is generated only by those p - and q -variables p γ n , q γ n correspondingto Reeb orbits which are multiple covers of orbits γ ∈ Γ. For a rigid immersedholomorphic curve v with asymptotic orbits Γ ′ + and Γ ′− let L Γ ′ + , Γ ′− be the spaceof formal power series in the variables p + γ n + with γ + ∈ Γ ′ + with coefficients whichare polynomials in the variables q − γ n − , γ − ∈ Γ ′− . Furthermore we introduce as in[EGH] the bigger space ˆ L Γ ′ + , Γ ′− whose elements are power series in p + γ n + and p − γ n − which are polynomials in q + γ n + and q − γ n − .While in standard SFT one collects the information about all moduli spacesof holomorphic curves in X by defining a potential f , we now define a local SFTpotential f v ∈ L Γ ′ + , Γ ′− counting only multiple covers of the fixed rigid immersedcurve with elliptic orbits v : S → X as in ([EGH], section 2.3.2). Definition 1.7.
For every choice of obstruction bundle sections (¯ ν ) coherentlyconnecting the coherent collections of obstruction bundle sections (¯ ν ± ) chosen forall positive and negative asymptotic Reeb orbits γ ± ∈ Γ ′± of v , we define the localSFT potential of a rigid immersed holomorphic curve v with elliptic orbits by f v ! = f (¯ ν ) v = X Γ + , Γ − s + ! s − ! κ Γ + κ Γ − M ¯ νv,d (Γ + , Γ − ) q Γ − − p Γ + + , where M ¯ νv,d (Γ + , Γ − ) = ¯ ν − (0) ⊂ M v,d (Γ + , Γ − ) . For the rest of this section we want to discuss in how far the local SFT Hamilto-nians h γ and the local SFT potentials f v depend on the collections of obstructionbundle sections (¯ ν ) needed to define them. While the above result about the localSFT Hamiltonian makes identities like the master equation { h γ , h γ } = 0 trivial,we can use it to prove the following important result Theorem 1.8.
Assume that the coherent collections of sections (¯ ν ± ) are fixed forall asymptotic Reeb orbits γ ± ∈ Γ ′± of v . Then the local SFT potential f v = f (¯ ν ) v of v is independent of the chosen collection of sections (¯ ν ) coherently connecting (¯ ν + ) and (¯ ν − ) . In particular, the algebraic count of multiple covers of the immersedcurve with elliptic orbits v is well-defined.Proof. For two collections of sections (¯ ν ) and (¯ ν ) coherently connecting (¯ ν + )and (¯ ν − ) let (¯ ν s ), s ∈ [0 ,
1] be the family of coherently connecting collections ofsections given by ¯ ν s = (1 − s ) · ¯ ν + s · ¯ ν , which defines a transversal section ¯ ν inCoker ¯ ∂ J over M × [0 , f sv = f (¯ ν s ) v denotethe corresponding family of local SFT potentials of v . Then it follows from theproof of the corresponding result for the standard SFT potential in ([EGH], section2.4) that f v and f v are homotopic through the homotopy f sv in the sense of [EGH],i.e., there exists another family k sv such that the family s f sv satisfies the followingHamilton-Jacobi equation in ˆ L Γ ′ + , Γ ′− , ∂ f sv ( p + , q − ) ∂s = G v (cid:16) p + , ∂ f sv ( p + , q − ) ∂p + , ∂ f sv ( p + , q − ) ∂q − , q − (cid:17) , ocal SFT 13 where G v ( p + , q + , p − , q − ) = { h Γ ′ + − h Γ ′− , k sv } = X γ ± ∈ Γ ′± κ γ − ∂ h Γ ′− ( p − , q − ) ∂p − γ − ∂k sv ( p + , q − ) ∂q − γ − + κ γ + ∂k sv ( p + , q − ) ∂p + γ + ∂ h Γ ′ + ( p + , q + ) ∂q + γ + . Since h Γ ′± = P γ ± ∈ Γ ′± h γ ± = 0, as by the above theorem h γ ± = h (¯ ν ± ) γ ± = 0 forall closed Reeb orbits γ ± and all coherent collections of sections (¯ ν ± ), it followsthat G v = 0, so that f sv must be independent of s ∈ [0 , f v = f v ∈L Γ ′ + , Γ ′− . (cid:3) We end this subsection by discussing how the local SFT potential f v dependson the choice of coherent collections of sections (¯ ν ± ) for all closed Reeb orbits γ ± ∈ Γ ′± , where it will turn out that f v indeed depends on this choice. Let L − , L + , L be generated by ( q − , p )-, ( q, p + )- and ( q − , p + )-variables, respectively.Following ([EGH], section 2.5) we define the operation L − × L + → L by( f − f + )( q − , p + ) = ( f − ( q − , p ) + f + ( q, p + ) − X γ κ − γ q γ p γ ) | L for f ± ∈ L ± , where L is the Lagrangian in the symplectic super-space spanned by( q − , p + )-variables which is determined by the equations q γ = κ γ ∂f − ∂p γ , p γ = κ γ ∂f + ∂q γ . For chosen homotopies (¯ ν +01 ), (¯ ν − ) from (¯ ν +0 ) to (¯ ν +1 ), (¯ ν − ) to (¯ ν − ), respectively,let f ′ + = f (¯ ν +01 )Γ ′ + ∈ , f ′− = f (¯ ν − )Γ ′− be the local SFT potential of the union of orbitcylinders in the cylindrical manifold equipped with non-cylindrical data. Note thatit again follows from the above theorem that the local SFT potentials f ′ + and f ′− are independent of the chosen homotopies. With this we can describe the change ofthe local SFT potential f v under different choices of coherent collections of sections(¯ ν ± ) as follows. Theorem 1.9.
For two different choices of coherent collections of sections (¯ ν ± ) and (¯ ν ± ) we denote by f v = f (¯ ν ± ) v , f v = f (¯ ν ± ) v ∈ L Γ ′ + , Γ ′− . Then we have f v = f ′− f v f ′ + . Note that this theorem follows by combining the algebraic formalism for com-position of cobordisms in ([EGH], section 2.5) with our above result stating that f v is independent of the chosen collection of sections coherently connecting two co-herent collections of sections. On the other hand, we want to emphasize that fromthe above result one can deduce as in [EGH] a functoriality as known from Floerhomology. To this end, observe that the local SFT potential f v ∈ L Γ ′ + , Γ ′− defines aLagrangian L f v in the symplectic super-space spanned by the p ± - and q ± -variables, L f v = { q + = ∂ f v ∂p + , p − = ∂ f v ∂q − } . Viewing functions in P ± Γ ± in the natural way as elements in the bigger space ˜ L Γ ′ + , Γ ′− we follow [EGH] and define maps f ± v : P Γ ′± → L Γ ′ + , Γ ′− , g g | L f v . It is shown in [EGH] that the local SFT potentials f ′ + and f ′− define not only au-tomorphisms of P Γ ′ + and P Γ ′− , respectively, but also an automorphism of L Γ ′ + , Γ ′− .We get the following functorial property of the maps f ± v . Corollary 1.10.
After applying the automorphisms of P Γ ′ + , P Γ ′− and L Γ ′ + , Γ ′− induced by f ′ + and f ′− , the map f , ± v : P Γ ′± → L Γ ′ + , Γ ′− gets replaced by the map f , ± v : P Γ ′± → L Γ ′ + , Γ ′− . Application: Stable hypersurfaces intersecting exceptionalspheres
Instead of discussing the full TQFT picture involving splitting and gluing ofthe underlying immersed curves with elliptic orbits, in this section we will showhow local SFT methods can be applied to embedding problems in symplecticgeometry. More precisely, we will show that every stable hypersurface whichintersects an exceptional sphere in a closed four-dimensional symplectic manifoldin a homological nontrivial way must carry an elliptic orbit.2.1.
Additional marked points and gravitational descendants.
For this weuse that a closed rigid nicely-embedded curve v : ( S , i ) → ( X, J ) is an exceptionalsphere . Indeed, it follows from the definitions that ind( v ) = 0 and δ ( v ) = 0, so that2 c N ( v ) = ind v − g + = − v ] · [ v ] = i ( v, v ) = 2 δ ( v ) + c N ( v ) = −
1. On the other hand, since c ( v ) = c ( v ∗ T X ) = c ( N v ) + c ( T S ) = c N ( v ) + 2 = 1 we get that the Fredholmindex of a d -fold multiple cover u = v ◦ ϕ in the moduli space M v,d is given byind( u ) = − c ( u ) = − d = 2( d −
1) and hence strictly positive for d > u in M v,d factor through v , u = v ◦ ϕ , it followsthat we can only expect to get non-zero integrals when the degree of the form istwo or less. Since adding one additional marked point enlarges the dimension ofthe moduli space by two, it follows that this way we will do not get contributionsfrom higher-dimensional moduli spaces. We solve this problem by additionallyintroducing gravitational descendants.Let M v,d,r denote the moduli space of d -fold coverings u = v ◦ ϕ carrying r additional marked points. In order to save notation, instead of integrating thepull-back of the canonical two-form on the sphere over the moduli space, wedirectly want to assume that every additional marked point z i gets mapped toa special marked point p i on the exceptional sphere under the covering map ϕ .Note that as an immediate consequence of the divisor equation in Gromov-Wittentheory we get that M v,d,r is given by d r copies of the moduli space M v,d = M v,d, without additional marked points, where d r is the number of preimages of thespecial points p , . . . , p r under the d -fold covering ϕ .On the other hand, with the help of the additional marked points we canintroduce r tautological line bundles L , . . . , L r over each moduli space M v,d,r .They are defined as the pull-back of the vertical cotangent line bundle of π i : M v,d,r +1 → M v,d,r under the canonical section σ i : M v,d,r → M v,d,r +1 mapping to the i -th marked point in the fibre. It follows that the fibre L i over ocal SFT 15 a smooth curve ( u, z , . . . , z r ) is given by the cotangent line to the underlyingRiemann sphere at the i .th marked point, ( L i ) (( u,z ,...,z r ) = T z i S . With thisthe local Gromov-Witten potential can be enriched by integrating products ψ j ∧ . . . ∧ ψ j r r of powers of the first Chern classes ψ i = c ( L i ), i = 1 , . . . , r over themoduli spaces M v,d,r . It follows from the work in [OP] for the case when the targetmanifold is a complex curve that the divisor M ( j ,...,j r ) v,d,r ⊂ M v,d,r Poincare-dual to ψ j ∧ . . . ∧ ψ j r r has a geometric interpretation in terms of branching conditions.2.2. Obstruction bundle = normal bundle using topological recursion.
Instead of discussing the general statement, from now on let us restrict to thesimplest non-trivial case d = 2, r = 1, j = 1. Here it follows that the submodulispace M v, , ⊂ M v, , consists of two-fold coverings u = v ◦ ϕ with one markedpoint mapping to the special point on the exceptional sphere which is additionallyrequired to be a branch point of ϕ . While the (real) dimension of the (unperturbed)moduli space is two which accounts for the second branch point of the coveringmap ϕ : S → S , the expected dimension of the moduli space M v, , given bythe Fredholm index is 2(2 − − ∂ J over M v, , of rank two.Since our curves have no punctures and hence there is no codimension-one bound-ary of the moduli space, it follows that the count of elements in the resultingperturbed moduli space ( M v, , ) ¯ ν = ¯ ν − (0) ⊂ M v, , is independent of the cho-sen section ¯ ν in Coker ¯ ∂ J . Using topological recursion relations for descendantsin Gromov-Witten theory we will show the count of elements M , ¯ νv, , is (up toa combinatorical factor coming from the divisor equation) given by homologicalself-intersection number of the exceptional sphere. Theorem 2.1.
We have M v, , ) ¯ ν = − .Proof. The idea of the proof is that, using the topological recursion relations ofGromov-Witten theory, we can relate the above moduli space to the moduli spaceof doubly-covered spheres with one node. Since both components need to be simply-covered spheres which are automatically regular, the transversality problem in theBanach space bundle localizes on the nodal coincidence relation and hence reducesto geometric transversality. Since we need three marked points to apply topologicalrecursion relations, we first apply the divisor equation twice to get M v, , ) ¯ ν = · M (1 , , v, , ) ¯ ν . Applying topological recursion relations we get that M (1 , , , ¯ νv, , = M v, , × ev M v, , ) ¯ ν = M v, , × ev M v, , ) ¯ ν where M v, , × ev M v, , = { (( u , w ) , ( u , w )) : u ( w ) = u ( w ) } and the second equality follows by applying the divisor equation in the reversedirection. Since we can assume after applying an automorphism of the domainthat the simple covering ϕ : S → S in u = v ◦ ϕ is the identity and hence u = v , note that we can identify M v, , and hence also M v, , × ev M v, , with S via ( u, z ) =: z . We now want to show that the obstruction bundle over M v, , × ev M v, , ∼ = S is given by the normal bundle to the exceptional sphere. Claim:
Coker ¯ ∂ J ∼ = N v . For this we make use of the fact that by proving transversality of the Cauchy-Riemann operator in the Banach space bundle
E ⊕ E over the Banach submanifold
B × ev B = { ( u , w ) , ( u , w ) : u ( w ) = u ( w ) } ⊂ B × B containing M × ev M ( M = M v, , ) we do not only get transversality for theCauchy-Riemann operator in E ⊕ E over the Banach manifold
B × B of discon-nected curves, but we also get that the evaluation map ev :
M × M → X × X ,(( u , w ) , ( u , w )) ( u ( w ) , u ( w )) is transversal to the diagonal in X × X (see[F3] for a proof of this lemma). It follows that the fibre of the cokernel bundle at z ∈ S ∼ = M× ev M is given by (Coker ¯ ∂ J ) z = coker D Nz , where D Nz is the restrictionof the componentwise linearization in the normal direction D N : H ,p ( N v ) ⊕ H ,p ( N v ) → L p (Λ , ⊗ i,J N v ) ⊕ L p (Λ , ⊗ i,J N v )to the subspace { ( ξ , ξ ) ∈ H ,p ( N v ) ⊕ H ,p ( N v ) : ξ ( z ) = ξ ( z ) } . Since bothcomponents are simple and hence regular, and hence D N is an isomorphism, itfollows thatcoker D Nz ∼ = H ,p ( N v ) ⊕ H ,p ( N v ) { ( ξ , ξ ) ∈ H ,p ( N v ) ⊕ H ,p ( N v ) : ξ ( z ) = ξ ( z ) }∼ = ( N v ) z ⊕ ( N v ) z ∆ ∼ = ( N v ) z . Putting everything together we get
M × ev M ) ¯ ν = Z M× ev M e (Coker ¯ ∂ J ) = Z S e ( N v ) = [ v ] · [ v ] = − . (cid:3) Equations for the local SFT potentials.
We now want to turn againfrom local Gromov-Witten theory to local SFT. To this end we consider a stablehypersurface V in the symplectic manifold X which intersects the exceptionalsphere Σ := v ( S ). Assuming that this intersection is homologically non-trivialin the sense that the union of circles C = Σ ∩ V defines a non-zero class in H ( V ), it follows that after neck-stretching along V (see [BEHWZ]) the closedholomorphic sphere v breaks up into two punctured holomorphic curves v + and v − (possibly with several connected components), connected by a collection Γ ofclosed Reeb orbits on V in the sense that Γ is the set of the negative or posi-tive asymptotic orbits of the punctured holomorphic curves v + and v − , respectively.For notational simplicity let us assume that Γ just consists of a single closedReeb orbit γ and V is separating, X = X + ∪ V X − , V = ∓ X ± . Then theholomorphic sphere v : ( S , i ) → ( X, J ) breaks up into two holomorphic planes v ± : ( C , i ) → ( X ± , J ± ). Note that we continue not to distinguish betweenthe compact symplectic manifolds with boundary X ± and their completions X ± ∪ R ∓ × V which are symplectic manifolds with cylindrical ends in the sense of[BEHWZ]. On the other hand, since ind( v ) = 0, we get from index additivity andregularity that ind( v + ) = ind( v − ) = 0.For the moment let us further assume that v + and v − are again immersed curveswith elliptic orbits and that γ is elliptic. We want to use our computation forthe local Gromov-Witten potential of v to prove results about the moduli spaces M v + , ( ∅ , Γ) and M v − , (Γ , ∅ ) from local SFT, where it turns out that the resultdepends on the behaviour of the Conley-Zehnder index for the multiple coveredorbits γ k . Recall that for every elliptic orbit γ there exists an irrational number θ such that for the Conley-Zehnder indices we have CZ( γ k ) = 2[ kθ ] + 1, where [ x ] ocal SFT 17 denotes the largest integer less or equal than x . It follows that CZ( γ ) − γ ) =2([2 θ ] − θ ]) − ∈ {− , +1 } . Introducing additional marked points and gravita-tional descendants as in local Gromov-Witten theory using branching conditions todefine moduli spaces M v + , , ( ∅ , Γ) and M v − , , (Γ , ∅ ), we now prove the followingtheorem. Theorem 2.2. If CZ( γ ) − γ ) = − then M ¯ νv − , ( γ , ∅ ) + M v + , , ) ¯ ν ( ∅ , ( γ, γ )) = M v − , , ) ¯ ν (( γ, γ ) , ∅ ) = −
14 ; if CZ( γ ) − γ ) = +1 then M ¯ νv + , ( ∅ , γ ) + M v − , , ) ¯ ν (( γ, γ ) , ∅ ) = M v + , , ) ¯ ν ( ∅ , ( γ, γ )) = − . In particular, while the summands on the left side depend on the choice of coherentobstruction bundle sections (¯ ν ) for γ , the sum is independent of this choice.Proof. Let u n = v ◦ ϕ n be a sequence of multiple covers of the exceptional sphere.After neck-stretching along the hypersurface V it follows from the compactnessresult in [BEHWZ] that a subsequence converges to broken holomorphic curve( u + , u − ), which are multiple covers of the holomorphic planes v + , v − , respectively, u ± = v ± ◦ ϕ ± . It follows that via compactness and gluing the moduli space M v, is related to the union of moduli spaces S Γ M v + , ( ∅ , Γ) × M v − , (Γ , ∅ )of possibly disconnected curves, where Γ = ( γ, γ ) or Γ = γ . It follows that,depending on whether we choose the special point on Σ = v ( S ) on Σ + = Σ ∩ X + or Σ − = Σ ∩ X − , we get that M v, , is related to the union of moduli spaces S Γ M v + , , ( ∅ , Γ) × M v − , (Γ , ∅ ) or S Γ M v + , ( ∅ , Γ) × M v − , , (Γ , ∅ ).In the case when Γ = ( γ, γ ) note that the curves u + in M v + , ( ∅ , ( γ, γ )) and u − in M v − , (( γ, γ ) , ∅ ) are either cylinders with two negative or positive punctures orpairs of two simple holomorphic planes. Since in the latter curves do not carrybranch points, it follows that the curves in M v + , , ( ∅ , ( γ, γ )) and M v − , , (( γ, γ ) , ∅ )are cylinders, so that the corresponding moduli spaces M v − , (( γ, γ ) , ∅ ) and M v + , ( ∅ , ( γ, γ )) must consist of pairs of simple holomorphic planes. While thelatter are automatically regular, it follows from the index and dimension additivitythat there is an obstruction bundle of rank two over the two-dimensional modulispaces M v + , , ( ∅ , ( γ, γ )) and M v − , , (( γ, γ ) , ∅ ), where the two (real) dimensionsagain account for the second branch point. Since the count of regular curvesis clear, it follows that the algebraic count of these moduli spaces is given by M v + , , ) ¯ ν ( ∅ , ( γ, γ )) and M v − , , ) ¯ ν (( γ, γ ) , ∅ ), respectively.In the case when Γ = γ , it follows that the curves u + in M v + , ( ∅ , γ ) and u − in M v − , ( γ , ∅ ) are holomorphic planes. When CZ( γ ) − γ ) = − u + ) = 2 and ind( u − ) = 0; similarly, when CZ( γ ) − γ ) = +1 thenind( u + ) = 0 and ind( u − ) = 2. Since dim M v + , ( ∅ , γ ) = dim M v − , ( γ , ∅ ) = 2 anddim M v + , , ( ∅ , γ ) = dim M v − , , ( γ , ∅ ) = 0, it follows from dimension reasonsthat in the first case we only get contributions from moduli spaces M v + , , ( ∅ , γ ) ×M v − , ( γ , ∅ ), while in the second case we only get contributions from mod-uli spaces M v + , ( ∅ , γ ) × M v − , , ( γ , ∅ ). While the curves in M v + , , ( ∅ , γ )and M v − , , ( γ , ∅ ) are automatically regular, it follows that we have obstructionbundles of rank two over the moduli spaces M v + , ( ∅ , γ ) and M v − , ( γ , ∅ ), so that the algebraic count of the moduli spaces is given by M v + , ) ¯ ν ( ∅ , γ ) or M v − , ) ¯ ν ( γ , ∅ ), respectively. (cid:3) Exceptional spheres cannot break along hyperbolic orbits.
In thisfinal subsection we want to show how the general idea of local symplectic fieldtheory can be used to prove new results about contact hypersurfaces in symplecticfour-manifolds. We emphasize that for the proof of the following statement we do not need to exclude odd hyperbolic a priori, since for our result we will only needto study multiple covers of the exceptional sphere and over cylinders over orbitsappearing in the neck stretching process.
Theorem 2.3.
Assume that the exceptional sphere splits after neck-stretching alongthe unit cotangent bundle of an oriented Lagrangian into punctured holomorphiccurves connected by a collection of closed Reeb orbits Γ in V which are all Morsenondegenerate. Then at least one of the orbits in Γ must be elliptic.Proof. After neck-stretching along the contact hypersurface, it is shown in[BEHWZ] that the exceptional sphere breaks along a finite collection Γ of closedReeb orbits on the contact hypersurface. We now want to consider the case wherethe special point is chosen to lie on the intersection locus C = Σ ∩ V = ∓ ∂ Σ ± of the exceptional sphere with the stable hypersurface. In this case it followsthat, after neck-stretching, the special marked point now lies on one of theorbit cylinders R × γ in R × V for γ ∈ Γ. The corresponding moduli spaces M γ, , (Γ + , Γ − ) of multiple covers appearing in the boundary of M v, , consists ofbranched covers of the orbit cylinder with one additional marked point which isrequired to be a branch point and mapped the special point (0 ,
0) on R × S ∼ = R × γ .Now assume that γ is hyperbolic. Using the additivity of the Conley-Zehnderindex for hyperbolic orbits, it follows that the Fredholm index of a curve in M γ,d, (Γ + , Γ − ) is determined by the Euler characteristic of the underlyingpunctured curve. While the virtual dimension as expected by the Fredholm indexcontinues to be zero for M γ, , (( γ, γ ) , ( γ, γ )), it is strictly negative for the modulispaces M γ, , (( γ, γ ) , γ ), M γ, , ( γ , ( γ, γ )) and M γ, , ( γ , γ ). It follows that,after choosing sections in the obstruction bundles over the latter moduli spaces oforbit curves to perturb the Cauchy-Riemann operator, we only need to care aboutthe case when Γ + = Γ − = ( γ, γ ).Since now both branch points sit over the orbit cylinder of γ , it follows thatwe have an obstruction bundle of rank two over the two-dimensional moduli space M γ, , (( γ, γ ) , ( γ, γ )), while there is no obstruction bundle for the multiple coversof the other components. While the latter already ensures that we do not need toexclude odd hyperbolic orbits apriori, we now even show that, after perturbing themultiple covers of the orbit cylinder, the latter multi-floor curves cannot contribute.For this we use that the corresponding generating function h γ, , counting perturbedholomorphic curves in the moduli spaces ( M γ,d, ) ¯ ν (Γ + , Γ − ), was computed in [F3].Since, for any choice of cokernel bundle sections making the moduli spaces regular,we have h γ, = 0 if γ is hyperbolic, it follows that there cannot be a nonzerocount of perturbed holomorphic spheres in M v, , when all breaking orbits in Γare hyperbolic. Since the latter contradicts our direct computation from above, wefind that at least one orbit in Γ must be elliptic. (cid:3) As an immediate corollary, we find an independent proof of the following resultfrom ([Wel], theorem 1.3). ocal SFT 19
Corollary 2.4.
Assume that a closed oriented Lagrangian surface L in a closedsymplectic four-manifold has a homologically nontrivial intersection with an excep-tional sphere Σ . Then L must be diffeomorphic to S or S × S .Proof. Here it suffices to observe that any surface of genus greater than one admits ametric (the uniformizing one) where all closed geodesics are hyperbolic and Morse.Since the same holds true for the corresponding closed Reeb orbits on its unitcotangent bundle, it directly follows from our result that the unit cotangent bundle(for the uniformizing metric) and hence the Lagrangian itself (with the propertiesstated above) cannot exist. (cid:3)
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