An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves
AAN INTRODUCTION TO HOMOLOGICAL MIRROR SYMMETRYAND THE CASE OF ELLIPTIC CURVES
ANDREW PORT
Abstract.
Here we carefully construct an equivalence between the derived category of coherentsheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the mostaccessible case of homological mirror symmetry. We also provide introductory background on thegeneral Calabi-Yau case of The Homological Mirror Symmetry Conjecture.
Contents
0. Introduction 21. Complex, Symplectic and K¨ahler manifolds 41.1. Complex manifolds 41.2. Almost Complex Structures 51.3. Symplectic Geometry 51.4. Lagrangian submanifolds and Hamiltonian flows 71.5. K¨ahler Geometry 81.6. Calabi-Yau Manifolds 82. Complex and Holomorphic Vector Bundles 92.1. Complex Vector Bundles 92.2. Connections on Vector Bundles 102.3. Holomorphic Vector Bundles 112.4. Sections of a holomorphic vector bundles 123. A Review of Elliptic Curves 143.1. Complex structures on Elliptic curves 143.2. The Calabi-Yau Structure of an Elliptic curve 144. Holomorphic Line Bundles on Elliptic Curves 154.1. Theta Functions 154.2. The Category of Holomorphic Line Bundles on Elliptic curves 175. The A-side 185.1. B-fields and the K¨ahler Moduli Space 185.2. The Fukaya Category 195.3. The Objects 205.4. The Morphisms 205.5. The A ∞ -structure 216. The Simplest Example 237. The B-side 277.1. The Derived Category of Coherent Sheaves 277.2. The Classification of Holomorphic Vector Bundles on E τ Date : September 16, 2012. a r X i v : . [ m a t h . S G ] J a n ANDREW PORT D b ( E τ ) and F K ( E τ ) 338.3. Mirror Objects 348.4. Morphisms of Vector Bundles 348.5. Morphisms of Torsion Sheaves 35Appendix A. Homological Algebra Background 36A.1. Chain Complexes 36A.2. The Derived Category 37Appendix B. Minimal, Calibrated, and Special Lagrangian Submanifolds 39B.1. Minimal Submanifolds 39B.2. Calibrated Submanifolds 40B.3. Special Lagrangian Submanifolds 40B.4. Graded Lagrangian Submanifolds 41References 420. Introduction
The concept of mirror symmetry , in its earliest forms, was conceived in the late 1980s byphysicists who observed that topologically distinct manifolds could give rise to equivalentquantum field theories and thus equivalent notions of physics. In 1994, roughly fifteenyears after these first observations , Kontsevich proposed a mathematically rigorous frame-work for this symmetry based on what is now known as The Homological Mirror SymmetryConjecture [29]. The principle conjecture made in [29] can be thought of as a definition ofwhat it means to be the “mirror dual” to a Calabi-Yau manifold. This definition (slightlyweakened from its original form) can be stated as follows. Let X be a Calabi-Yau manifold. A complex algebraic manifold, ˜ X , is said to be mirrordual to X if the bounded derived category of coherent sheaves on ˜ X is equivalent to thebounded derived category constructed from the Fukaya category of X “(or a suitable enlarge-ment of it)”. The (derived) Fukaya category of X is built from only the K¨ahler structure (coupledwith a B-field) of X , whereas the (derived) category of coherent sheaves on ˜ X depends ononly the complex structure of ˜ X . In particular, homological mirror symmetry (HMS) is arelationship between symplectic and algebraic geometry. Currently it is “widely believed”([15] section 1.4) that to each Calabi-Yau manifold, such a mirror dual exists and is alsoCalabi-Yau.In the case of Calabi-Yau manifolds, as it is stated above, this equivalence (or a closeversion of it) has been proven completely in the cases of elliptic curves [37] and (for X a) N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 3 quartic surface [38]. Aspects of the conjecture have been proven in other cases, particularlyin the cases of abelian varieties and Lagrangian torus fibrations[14, 30]. There are alsoconjectured extensions of HMS to non-Calabi-Yau manifolds (e.g. [2, 26]), where the mirrordual of a symplectic manifold is taken to be a pair ( ˜ X, W ) of a complex variety ˜ X and aholomorphic function W . Complete proofs of this more general HMS exist for all surfacesof genus g ≥ In this thesis we will concentrate on the Calabi-Yau case of HMS. We will give an in-troduction to the subject through the example where X is an elliptic curve. This case isparticularly accessible thanks to the fact that elliptic curves are particularly well-understoodgeometric objects and the intention of this document will be to provide enough detail (pre-sented at an introductory enough level) for a geometrically-oriented mathematician to attaina good understanding of this case and its more general underlying conjecture (without thesubstantial and diverse prerequisites typically associated with HMS). We will pay particu-lar attention to explanations complementary to those included in [37, 31] and we will makeeffort to give our explanation in terms of the general Calabi-Yau case whenever it is possibleto do so without straying too far from our goal of understanding the case of elliptic curves.In the ICM talk [29] where Kontsevich gave his original HMS conjecture, he also gave arough comparison of the two sides of this categorical equivalence in the case of elliptic curves.About five years later Polishchuk and Zaslow gave a proof of this case in a weakened con-text, which we will describe in this thesis. To summarize quickly, let τ = b + iA ∈ R ⊕ i R > ,and let E τ denote the smooth 2-torus equipped with the symplectic form Adx ∧ dy andanother 2-form bdx ∧ dy called the “B-field”. The mirror dual of E τ is the complex 1-torus E τ := C / Λ given by the natural action of lattice Λ = < , τ > . This “weakenedcontext” mentioned above is an equivalence between zeroth cohomology of the Fukaya cate-gory, H ( F uk ( E τ )), (with biproducts formally added) and the derived category of coherentsheaves, D b Coh( E τ ). The functor between these categories will be constructed from a sur-prisingly strange relation sending closed geodesics of E τ equipped with flat complex vectorbundles (and some additional structure) to holomorphic vector bundles and torsion sheaves.In fact the indecomposable torsion sheaves will be the images of the vertical geodesics (i.e.projections of vertical lines through the quotient R / Z ). The image of morphisms will bedescribed in terms of a relation sending intersection points (of geodesics) to theta func-tions! Perhaps most importantly, as the composition of morphisms in the Fukaya categoryis given by formal sums of intersection points with coefficients determined from countingpseudo-holomorphic disks (marked at the intersection points), the functor will encode thisenumerative information in (compositions of morphisms of certain coherent sheaves, whichcan be described in terms of) linear combinations of theta functions (with constant matrix X, W ) is called a
Landau-Ginzberg model and the function W is called a Landau-Ginzberg superpotential .6It is possible a complete proof of the genus zero case also exists or is implicit in related works (see [3] for an overviewof this case).7The author apologizes any complete proofs he is unaware of and also for the many partial results not included in theabove references for both the Calabi-Yau and non-Calabi-Yau cases.8Elliptic curves (i.e. one-dimensional complex tori) are the only one-dimensional Calabi-Yau manifolds.9To be specific, this will be an equivalence between additive categories. Generally speaking, the Fukaya category isnot a true category but instead an A ∞ -category (or worse a curved A ∞ -category) only associative at the level ofcohomologies. In this one-dimensional case, however, the obstruction to its associativity vanishes and, after formallyadding biproducts, we can construct an equivalence with the derived category of coherent sheaves without consideringthe enlargement to the derived category constructed from the Fukaya category. ANDREW PORT coefficients).This thesis is organized as follow. In section 1 we give a brief review of the necessarybackground/definitions from symplectic geometry and complex geometry, then in section2 we continue to a brief discussion of the necessary background from the theory of com-plex/holomorphic vector bundles. Sections 3 and 4 given detailed constructions all necessarygeometric structures and then all holomorphic line bundles on elliptic curves. Section 5 and6 discuss the Fukaya category and together give a detailed exposition of [37]’s “simplestexample” of the simplest example of HMS. In sections 7 and 8 we then discuss the de-rived category of coherent sheaves on an elliptic curve and construct the above mentionedfunctor between this category and an enlargement of H ( F uk ( E τ )). We also include twoappendices, the first giving a minimalist review of general derived categories and the secondgiving an introduction to the topic of special Lagrangian manifolds and related geometries. Acknowledgement.
I’d like to thank my thesis committee members, Jerry Kaminker,Motohico Mulase, and Andrew Waldron. Also I’d like to thank the many professors andother graduate students who’ve spent time teaching me mathematics. In particular AndrewWaldron and David Cherney, who spent countless hours attempting to teach me quantumfield theory, and Michael Penkava, who first introduced me to the homological viewpoint ofmirror symmetry. Lastly I’d like to again thank thank my advisor, Motohico Mulase, who’sput a great deal of effort into developing me as a mathematician.1.
Complex, Symplectic and K¨ahler manifolds
Here we will give some very basic review of complex, symplectic and K¨ahler manifolds.1.1.
Complex manifolds. A complex manifold is a smooth, 2 n -dimensional manifold, M , with an atlas charting M into C n such that the transition functions are holomorphic. Equivalently, M is a smoothmanifold which admits a complex structure (defined below). Example: C n is a complex manifold with one chart atlas given by the identity map. Example: CP n is a complex manifold with an atlas given by the maps φ i : [ t , ..., t i − , , t i +1 , ..., t n ] → ( t , ..., t i − , t i +1 , ...t n ).An important distinction between complex and smooth manifolds is that not all complexmanifolds are submanifolds of C m . In fact, by Liouville’s theorem, any compact connectedsubmanifold of C m is a point. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 5
Almost Complex Structures.
We can “complexify” any even dimensional vector space V by choosing a linear map J : V → V such that J = −
1. This gives V the structure of a complex vector space withmultiplication defined by ( a + ib ) · v = ( a + J b ) v .A smooth manifold M is said to be almost complex if it admits a smooth tangent bundleisomorphism J : T M → T M (called an almost complex structure ) that squares to the neg-ative identity. All complex (and symplectic) manifolds admit an almost complex structure,but the opposite is not true (for complex or symplectic).More specifically, on an almost complex manifold, (
M, J ), we can always find complexcoordinates z µ = x µ + iy µ : M → C such that at a given point, p ∈ M , we have J ∂∂x µ = i ∂∂y µ and J ∂∂y µ = − ∂∂x µ However, it is in general not possible to find coordinates such that this is true in anentire neighborhood of p . If we can find such coordinates in a neighborhood of every point p , then these glue together as a holomorphic atlas for M , and J is called a complex structure . Theorem 1.1 (The Newlander-Niremberger Theorem [35]) . An almost complex manifold ( M, J ) is a complex manifold with complex structure J if and only if the Nijenhuis tensorvanishes. I.e. N J ( v, w ) := [ v, w ] + J [ v, J w ] + J [ J v, w ] − [ J v, J w ] ≡ for all v, w ∈ Γ( T M ) , where [ · , · ] is the Lie bracket of vector fields over M .Example: All almost complex structures on a surface are integrable.
Remark 1.2.
All symplectic manifolds admit an almost complex structure. In fact, givena symplectic manifold (
M, ω ), choosing either an almost complex structure, J , or a Rie-mannian metric g , defines a “compatibility triple” ( ω, J, g ) such that g ( u, v ) = ω ( u, J v ). Definition 1.3.
Let f : M → M (cid:48) be a map between to almost complex manifolds ( M, J )and (
M, J (cid:48) ). f is said to be ( J, J (cid:48) )-holomorphic if df J = J (cid:48) f .1.3. Symplectic Geometry.
The study of symplectic geometry was originally motivated by the field of Hamiltonianmechanics, which sought to describe the laws of classical mechanics in terms of “conservedquantities” (see section 1.4) on a symplectic manifold representing the space of all possiblevalues of position and momentum for some system. The term symplectic was introducedby Weyl in 1939 (see the footnote on p. 165 of [43]) as a greek analog to the latin root of“complex” meaning “braded together”. Despite having such early origins, little was under-stood about symplectic geometry until Gromov introduced the idea of pseudo-holomorphiccurves in 1985. Since then the fields of Gromov-Witten invariants and Floer homologies
ANDREW PORT have greatly expanded our ability to understand the mathematical structure of these ge-ometries.
Definition 1.4. A symplectic manifold is a pair ( M, ω ) of a smooth manifold, M , and aclosed non-degenerate 2-form on M , ω . We call such an ω a symplectic form . Remark 1.5.
The non-degeneracy of this anti-symmetric form enforces that all symplecticmanifolds are even dimensional and at each p ∈ M there exists a basis e , ...e n , f , ..., f n of T p M such that ω ( e i , f j ) = δ ij and ω ( e i , e j ) = ω ( f i , f j ) = 0 for all i, j . Examples: • n -dimensional Euclidean space: R n with its standard coordinates and ω = (cid:80) ni =1 dx i ∧ dx i + n . • Any 2-surface with any non-vanishing 2-form. • The 2 n -torus : C n / Z n n -times (cid:122) (cid:125)(cid:124) (cid:123) ∼ = S × ... × S with its standard angular coordinates and (cid:80) ni =1 ω = dθ i ∧ dθ i + n • The cotangent bundle T ∗ M of any smooth n -dimensional manifold M can be givena symplectic structure. Letting ( x , ...x n ) denote local coordinates on M , we candefine local coordinates on T ∗ M by the map( x, ξ ) (cid:55)→ ( x , ..., x n , ξ , ..., ξ n ), where ξ i are the components of ξ with respect to thebasis local coordinate from, i.e. ξ = (cid:80) ξ i dx i . ω = (cid:80) dx i ∧ dξ i then definines a symplectic form on T ∗ M . Non-Examples: • Any odd dimensional or non-orientable manifold. • By Stoke’s theorem:
Any compact connected manifold with vanishing second coho-mology (e.g. S n for n > η denote its primitive, give rise to the exact volume form ω n = d ( η ∧ ω n − ). Theorem 1.6 (Darboux’s theorem) . Let ( M, ω ) be a n -dimensional symplectic manifoldand p ∈ M . Then there exist local coordinate chart ( U, ( x , ..., x n , y , ..., y n )) such that p ∈ U and, on U , ω = n (cid:88) i =1 dx i ∧ dy i Proof.
A proof of this theorem can be found in most introductory texts on symplecticgeometry (e.g. [9]). (cid:3) n -torus or, in the case n = 1,an elliptic curve N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 7
Remark 1.7.
This says that all symplectic manifolds locally all look alike. For purpose ofcomparison, any Riemannian metric can be made (by choice of coordinates) to look like theEuclidean at a single point, but not necessarily in an entire neighborhood. If we can findsuch a local coordinate patch around every point, then our manifold is, by definition, flat .1.4.
Lagrangian submanifolds and Hamiltonian flows.
The objects of the Fukaya category are closed Lagrangian submanifolds (with some addi-tional structure we will describe later). The submanifolds are usually either required to be“special” (in the sense described in appendix B) or else only considered up to “Hamiltonianequivalence” in the sense we will describe now.
Definition 1.8.
Let (
M, ω ) be a symplectic manifold. A submanifold L ⊂ M is called Lagrangian if ω | L ≡ V be any vector field on a smooth manifold M . We can locally (in time) solve forthe “flow” of a point x ∈ M in the direction of V (i.e. solve ˙ γ ( t ) = V γ ( t ) , γ (0) = x ). It issimple to show that this gives us a family of diffeomorphisms ψ t : M → M called the flow of V . We call V complete if its flow exists for all time. A vector field, V , on a symplecticmanifold, ( M, ω ), is called a symplectic if its flow preserves ω , i.e. ψ ∗ t ω = ω .By Cartan’s formula for the Lie derivative we have ddt ψ ∗ t ω = Lie V ω = d ( V (cid:120) ω ) + V (cid:120) dω = d ( V (cid:120) ω )Where we use (cid:120) to denote the interior product, i.e. V (cid:120) ω : W (cid:55)→ ω ( V, W ).This tells us that V is symplectic if and only if V (cid:120) ω is closed. If V (cid:120) ω is exact, thenthere exists some function H : M → R such that dH = V (cid:120) ω . Moreover, this function H ispreserved by the flow, i.e. ddt ψ ∗ t H = dH ( V ) = 0. In this case, ψ t is called a Hamiltonianflow , H a Hamiltonian function , and V , a Hamiltonian vector field .Notice that ω defines a perfect pairing between vectors and covectors. Thus given anyfunction H , we can find a vector field V such that dH = V (cid:120) ω . Definition 1.9.
Two Lagrangian submanifolds, L and L (cid:48) are said to be Hamiltonian equiv-alent if there exists a Hamiltonian flow ψ : I × M → M such that ψ t ( L ) = L (cid:48) for some t ∈ I . ANDREW PORT
K¨ahler Geometry.
A smoothly varying positive-definite hermitian inner product on the tangent bundle of analmost complex manifold is called a
Hermitian metric . A (resp. almost) complex manifoldequipped with such a structure is called a (almost) Hermitian manifold .An almost Hermitian manifold (
M, J, h ) has an associated Riemmanian metric and non-degenerate 2-form given by its symmetric and anti-symmetric parts, i.e. g = ( h + ¯ h ) and ω = i ( h − ¯ h ). Morover, as any smooth manifold admits a Riemannian metric g , the exis-tence of an almost complex structure J gives us the existence of a non-degenerate 2-form ω ( X, Y ) = g ( J X, Y ) and thus a Hermitian metric h = g − iω . Definition 1.10. A K¨ahler manifold is an almost Hermitian manifold that satisfies anintegrability condition that can be stated in the following three equivalent ways: • ω is closed and J is integrable • ∇ J = 0 • ∇ ω = 0where ∇ is the Levi-Cevita connection of g . Example:
All Riemann surfaces are K¨ahler.1.6.
Calabi-Yau Manifolds.
Studied by mathematicians since at least the 1950s [5, 6], Calabi-Yau manifolds have,since the 1980s [7], been of particular importance to physicists studying the subject of su-perstring theory. Superstring theory is a unified theory seeking to describe all elementaryparticles and fundamental forces of nature by modeling particles as vibrating strings. Forthe superstring model to be a consistent physical theory (e.g. to predict massless photons)it is necessary for space-time to be ten-dimensional. Experimental observation has led us tobelieve that our space-time locally looks like four-dimensional Minkowski space, M , (i.e. R equipped with the pseudo-Riemannian metric d(cid:126)x − dt ). Thus if these extra six dimen-sions do exist, it is expected that space-time locally look like the product M × X for X ,speaking informally, some compact space with dimensions of tiny length (on the order of thePlanck length). Different choices of X (and, of course, product structure) lead to different“effective” four-dimensional theories. It was shown in [7], that (under certain assumptionsabout the product structure of space-time) X must be taken to be a Calabi-Yau manifoldin order for this effictive four-dimensional theory to admit an N = 1 supersymmetry (aproperty popularly desired by physicists).We define an n -dimensional Calabi-Yau manifold to be a compact n -dimensional K¨ahlermanifold admitting a nowhere vanishing holomorphic n -form, which we call its Calabi-Yauform .This definition of Calabi-Yau implies that our K¨ahler structure has a vanishing firstChern class and thus can be thought of as Ricci-flat by “The Calabi Conjecture”, which we
N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 9 state here as proven by Yau [45, 46].
Theorem 1.11.
Let M be compact complex manifold admitting K¨ahler form ω and suppose ρ (cid:48) is some real, closed (1 , -form on M such that [ ρ (cid:48) ] = 2 πc ( M ) . Then there is a uniqueK¨ahler form ω (cid:48) on M such that [ ω ] = [ ω (cid:48) ] ∈ H ( M, R ) and ρ (cid:48) is the Ricci form of ω (cid:48) .In particular, if the first Chern class vanishes on M (w.r.t. a K¨ahler metric), then M admits a unique cohomologically equivalent Ricci-flat (i.e. ρ = 0 ) K¨ahler structure.Example: In one-dimension this implies that that our compact Riemann surface is paral-lelizable. Thus the only examples are those of genus one.2.
Complex and Holomorphic Vector Bundles
Studying HMS requires a good understanding of both complex and holomorphic vectorbundles. We will give some back review of these subjects here as relevant to this thesis.2.1.
Complex Vector Bundles.Definition 2.1.
A rank k complex vector bundle is a vector bundle C k → E π −→ M whosetransition functions are C -linear. Example: The trivial bundle , M × C k proj −−−→ M , is a rank k complex vector bundle. Thecomplexified tangent bundle of an almost complex manifold is also a complex vector bundle.These two examples are rarely the same. In fact, the complex (one) dimensional torus isthe only Riemann surface where this is true (for k = 1).Let us classify all complex vector bundles on the k -sphere. Lemma 2.2.
Every vector bundle over a contractible base is trivial.Proof.
If this were not true, our definition of vector bundle would not make sense. (cid:3)
Theorem 2.3.
There is a 1-1 equivalence between homotopy classes of functions f : S k − → GL k ( C ) and complex vector bundles on S k .Proof. Let E → S k be any rank n complex vector bundle. By the above lemma, E can becovered by two trivializations, ψ ± : π − ( D k ± ) → D k ± × C n , where D + ∩ D − ∼ = ( − (cid:15), (cid:15) ) × S k − .So the transition functions of this two chart atlas are defined by a homotopy f t : S k − → GL k ( C ) (for t ∈ ( − (cid:15), (cid:15) )). All that is left to do then is show this homotopy class is welldefined, i.e. if we take any two chart atlas, it gives us the same homotopy class. To seethis notice that we can contract φ ± to a constant map over a point. As GL n ( C ) is pathconnected, all such charts are homotopy equivalent. (cid:3) f t , is sometimes called a clutching function .12 GL n ( R ), on the other hand, has not one but two path connected components. Though we similarly classify all orientable real vector bundles on the k -sphere. Remark 2.4.
From this we have that all complex vector bundles on the circle are trivial(we can also see from the proof that we have two choices of real bundle structure - thecylinder or M¨obius band). The above technique can be used for any manifold that can becovered by two contractible charts. For example, all bundles on C ∗ are also trivial. Actually,we can generalize this (as is done in the finally of [4]) to manifolds having a “good” coverof r charts (i.e. manifolds of finite type) to find a bijection between rank k complex vectorbundles and homotopy classes of maps M → Gr k ( C n ) for any n ≥ rk . In fact, we canclassify complex vector bundles over an arbitrary manifold using the infinite Grassmannian(see [21]).Let E → M be a real rank k vector bundle over an n dimensional manifold M .If k > n , then there exists a nonvanishing global section of E . Similarly if E is complex andhas fiber C k , then there exists a global section if k > n/
2. In particular this tells us thenany vector bundle looks like E = E (cid:48) ⊕ I k − n where I is the trivial bundle and E (cid:48) is somebundle with an n dimensional fiber.Note that taking any section of E we can use the transversality theorem to assume it istransversal to the zero section and thus, locally, has finite zeros. Definition 2.5.
The degree of a complex vector bundle E is measured using as section σ which is transverse to the zero section. deg( E ) = (cid:80) ( − sgn ( det ( dσ p )) This is well-defined and equivalent to the degree homomorphism given by the first Chernclass (or the exponential sheaf sequence).2.2.
Connections on Vector Bundles.Definition 2.6.
Let G be a topological group and M a manifold. A principle G -bundle over M is a fiber bundle P π −→ M equipped with a fiber-preserving continuous right actionof G on P π −→ M which acts freely and transitively on its fibers. In other words, we have acontinuous right action of G , ρ : P × G → P , such that π ◦ ρ = π and at any point in p ∈ P ,the induced map ρ p : G → π − ( π ( p )) is a homeomorphism. Example:
Consider the frame bundle, GL ( E ), of some vector bundle F → E → M . GL ( E )is a principle GL ( F )-bundle under the action ( p · g )( v ) = p ( g · v ). This is a useful way tothink about vector bundles as is shown by the following theorem. Theorem 2.7.
The gauge equivalence classes of flat connections on a principle G -bundleover a connected manifold M are in one-to-one correspondence with the conjugacy classesof representations of π ( M ) → G .Proof. A simple proof of the fact (from the perspective of distributions) can be found insection 2.1 of [33]. (cid:3)
Remark 2.8.
The map π ( M ) → G is known as the monodromy representation of theconnection. The holonomy group of a connection (based at point x ∈ M ) is defined interms of the parallel transport operator, P , as Hol = {P γ | γ is a loop based at x } . The N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 11 restricted holonomy group , Hol is the normal subgroup given by contractible loops. Themonodromy representation is then the natural surjection π ( M ) → Hol/Hol . A connectionis flat if and only if Hol is trivial; thus in our case, this surjection is also sometimes calledthe holonomy homomorphism . Corollary 2.9.
The gauge equivalence classes of flat connections on rank k complex vectorbundles over S are in one-to-one correspondence with the conjugacy classes of GL k ( C ) . Proof. π ( S ) ∼ = Z and any representation Z → GL k ( C ) is determined by the image of .For this same reason, two representations are conjugate if and only if the images of theirgenerators are conjugate. (cid:3) Remark 2.10.
The image of is sometimes referred to (e.g. [31, 37]) as the “monodromyoperator” or simply the “monodromy” of the pair connection. In section 5, when discussingthe Fukaya category for an elliptic curve, we will restrict our attention to just flat connections(on vector bundles over S ) whose monodromy operators have only eigenvalues of unitmodulus.2.2.1. Local Systems. A sheaf of locally constant functions on a topological space, X , is mapping (satisfyingthe sheaf axioms) from the open sets of X to some fixed module. E.g. the space of con-stant functions { f : X → R | f ( x ) = a for all x ∈ X } forms such a sheaf when we considerall restrictions of these functions to open sets in X . A local system is the more generalconcept of a sheaf which locally looks like a sheaf of constant functions, but globally maybe twisted. For example, considering a flat vector bundle equipped with connection ∇ , thespace of local sections, s , that satisfy ∇ s = 0 form a local system. In mirror symmetryliterature the terms flat complex vector bundle (equipped with connection) and local system are often used synonymously.We will often use the term “local system” in this thesis to describe a Lagrangian submani-fold equipped with a complex vector bundle and a flat connection.2.3. Holomorphic Vector Bundles.Definition 2.11. A holomorphic vector bundle is a complex line bundle over a complexmanifold whose transition functions are holomorphic.Some examples are given by the canonical line bundle and the cotangent bundle of anycomplex manifold.It is important to note that, while every complex vector bundle over a Riemann surface ,admits a holomorphic structure, this structure is not unique. This does not complicateworking with line bundles much as the space of line bundles over a Riemann surface, C , GL k ( C ) are determined by the Jordan canonical form.14also often called a local coefficient system (of genus g ) forms a group, Pic( C ) ∼ = Z × J ( C ) where J ( C ) ∼ = C g / Z g denotes the space oftopologically trivial line bundles. The group structure of the Picard group , Pic( C ), is givenby the tensor product (and thus inversion by the dual space operator). The degree functiongives a homomorphism to Z with kernel, the Jacobian variety , J ( C ).In contrast to the complex case, holomorphic vector bundles (over Riemann surfaces) donot all split into sums of holomorphic line bundles. This makes holomorphic vector bundlesmuch more difficult to work with than their complex analogs. Luckily in the genus one casethese objects have a fairly simple classification discovered by Atiyah [1] (we will state hisresult in section 7).Our ability to use Atiyah’s classification in a black box manner will be largely aided bythe fact that we can pull all holomorphic bundles on C / Z ∼ = C ∗ / Z back to C ∗ where allvector bundles are trivial. Theorem 2.12.
All holomorphic vector bundles on a non-compact Riemann surface areholomorphically trivial.Proof.
For a simple complex analytic proof by induction on rank (see theorem 30.4 on p.229 of [12]). (cid:3)
We will also later make some minor use of the line bundle-divisor correspondence.A divisor on a Riemann surface, C , is a finite linear combination of points in C with integercoefficients. To divisors D and D (cid:48) are linearly equivalent if there is a meromorphic function, f , on C such that D − D (cid:48) = ord ( f ). A proof of the following fact can be found in anyintroduction to algebraic geometry or Riemann surface theory (e.g. [42]).
Theorem 2.13 (Line bundle-Divisor Correspondence) . There is a bijection between holo-morphic line bundles and divisors modulo linear equivalence.
Sections of a holomorphic vector bundles.
The degree of a complex vector bundle is a topological invariant, but is very useful forstudying holomorphic bundles. For example, if a holomorphic line bundle has a global holo-morphic section, then it must have a positive degree (by the Cauchy-Riemann equations).In particular, all global holomorphic sections must vanish at the same number of points,and this number, if nonzero, is equal to the degree. If the degree is negative, then theline bundle admits no global holomorphic sections. This can be seen by using the Cauchy-Riemann equations with definition 2.5.
Theorem 2.14 (Grothendieck’s Vanishing Theorem) . If V is a holomorphic vector bundleover an n -dimensional complex manifold, X , then H k ( X, V ) = 0 for k > n .Proof. A proof of this can be found in section III.2 of [19]. (cid:3) C ∗ as opposed to C as a convenience (we then only have to consider a Z action).17By ord ( f ) we mean the sum of all zeros and poles weighted by their orders. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 13
Remark 2.15.
This means on a Riemann surface, we know H k ( C, V ) = 0 unless k = 0 , H ( C, V ) = Γ( V ) (by the definition ofsheaf cohomology) and by the below Serre Duality, H ( C, V ) ∼ = H ( C, K C ⊗ V ∗ ). In theelliptic curve case, we then have that H ( V ) ∼ = Γ( V ∗ ) = Hom( V , O C ).It is also worth noting that H ( X, O ∗ X ) parameterizes holomorphic line bundles, where by O ∗ we mean the sheaf of non-vanishing holomorphic functions. A similar statement can bemade for real and complex line bundles. Theorem 2.16 (Riemann-Roch Theorem) . If V is a holomorphic vector bundle over aRiemann surface C , then χ ( C, V ) = deg( V ) + (1 − g ) rank ( V ) . Here χ is the Euler characteristic of V , defined as χ ( V ) := (cid:80) k ( − k h k ( V ) for h k ( V ) := dim( H k ( C, V )). Proof.
This particular statement is sometimes called “Weil’s Riemann-Roch theorem” andhas a generalization to higher dimensional compact complex manifolds called the Hirzebruch-Riemann-Roch theorem. A direct proof of this particular statement can be found on page65 of [17]. (cid:3)
Theorem 2.17 (Serre Duality) . If V is a holomorphic vector bundle over an n -dimensionalcompact complex manifold, X , then H k ( X, V ) ∼ = H n − k ( X, K X ⊗ V ∗ ) ∗ . Here K X denotes the canonical line bundle of X (i.e. K X := (cid:86) n, X ). Proof.
A proof of this statement of Serre duality, sometimes called “Kodaira-Serre duality”,can be found in section 4.1 of [25] or section 1.2 of [16]. (cid:3)
Remark 2.18.
Recall that on any Calabi-Yau manifold, K X is the trivial bundle. Com-bining these three theorems, we have, for any holomorphic vector bundle, V , over a genusone compact Riemann surface, C , deg( V ) = h ( V ) − h ( V ) = h ( V ) − h ( V ∗ ). In particular,if V is rank one, then deg( V ) = (cid:40) h ( V ) if deg( V ) > − h ( V ∗ ) if deg( V ) < A Review of Elliptic Curves
We define an elliptic curve to be the Riemann surface given by the quotient C / Λ and itsinherited complex structure, for Γ some Z lattice. All complex structures (up to biholo-morphic equivalence) on a smooth 2-torus are given in such a way. Choosing Λ = < , τ > ,we get a bijective relationship between elliptic curves and choices of τ with Im ( τ ) > E τ := C / < , τ > and will always assume,for reasons of convenience, Im ( τ ) > C are all preserved by translations.Thus they all descend to C / Λ. We will talk about this in more detail below, but for nowlet us just show we have a holomorphic atlas.3.1.
Complex structures on Elliptic curves.
Observing that the quotient map C π −→ C / Λ is open, we are given an holomorphic atlas (cid:83) z ∈ C { ( π ( B (cid:15) ( z )) , π | − π ( B (cid:15) ( z )) ) } where (cid:15) is some small number such that (cid:15) < | λ | for all λ ∈ Λ.Our transition functions then differ from the identity map by an element of Λ and thus areholomorphic.
Remark 3.1.
Sometimes we will find it convenient to reparameterize our curve as E τ = C ∗ / Z =: E q using the homomorphism u : C → C ∗ given by u : z (cid:55)→ e πiz so that thequotient is over multiplication by q = e πiτ . Throughout our discussion of HMS for theelliptic curve, τ and q will always be assumed to have this relationship, q = e πiτ .3.2. The Calabi-Yau Structure of an Elliptic curve.
As the standard Hermitian structure on C is invariant under translations, it descendsto a Hermitian structure on C / Λ. The Hermitian metric’s associated K¨ahler form is auto-matically closed (being a top form) and compatible with our descendant complex structure.Thus C / Λ is a K¨ahler manifold.Let ξ z = π | B (cid:15) ( z ) denote our local coordinate charts used above. Notice that dζ is well-defined globally; this tells us C / Λ is a Calabi-Yau Manifold.From theorem 1.11, we then know that given a K¨ahler structure on E τ , there is a uniquecohomologous Ricci-flat K¨ahler structure.As H ( E τ , R ) = R , we see that all Ricci-flat K¨ahler structures on E τ must be the positive scalar multiples of the one we’ve pulled down from C . Thus our K¨ahler structure on complexmanifold E τ is determined by its volume, which we will denote by A . A >
0, on the symplectic structure can be seeto mirror the convention that Im ( τ ) > N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 15
Choosing the K¨ahler structure on E τ given by a volume of A , we have: ω = vol g = Adx ∧ dy = i Adζ ∧ dζg = A ( dx ⊗ dy + dy ⊗ dx ) = − i A ( dζ ⊗ dζ − dζ ⊗ dζ )As E τ is compact, it has no non-constant holomorphic functions. Thus all CY forms areconstant multiples of dζ . This can of course be generalized to any Calabi-Yau manifold.To construct the Fukaya category, we will use the fact that our manifold is Calabi-Yau/K¨ahleroften, and sometimes it is even helpful to fix a Calabi-Yau form (e.g. see appendix B.4) orcomplex structure; the final result, however, will only depend on our choice of symplecticstructure and B-field (see section 5 for a definition of B-fields).4. Holomorphic Line Bundles on Elliptic Curves
Theta Functions. As E τ is compact, any holomorphic function, E τ → C , is constant. We can construct allmeromorphic functions on E τ in a similar fashion to how one would on projective space.In this analogy, the role of homogeneous polynomials are played by theta functions. Thatsaid, we are particularly interested in theta functions because they will give us bases forour B-side homsets.Consider the classical Jacobi theta function θ τ : C → C given by θ τ ( z ) := (cid:88) m ∈ Z e πi [ m τ +2 mz ] = (cid:88) m ∈ Z q m u m As we are always assuming Im ( τ ) >
0, this converges for all z ∈ C . Proposition 4.1. i) θ τ ( z + 1) = θ τ ( z )ii) θ τ ( z + τ ) = e − πi [ τ +2 z ] θ τ ( z ) = q − / u − θ q ( u )iii) θ τ ( z ) = 0 ⇐⇒ z = + τ + ( k + (cid:96)τ ) for k, (cid:96) ∈ Z (i.e. θ q ( u ) = 0 ⇐⇒ u = − q + (cid:96) )and these zeros are simple. iv) θ τ ( − z ) = θ τ ( z )v) θ τ ( τ − z ) = e πiz θ τ ( τ + z ) Proof. (i) and (iv) are trivially true.(ii) can be seen by a shift of the sum index (sending m → m − θ (cid:48) /θ around a fundamental parallelogram is equalto 1 / (2 πi ).(v) can be seen by combining (ii) and (iv). (cid:3) Letting θ ( x ) τ denote the translation θ τ with zeros at x + < , τ > (i.e. θ ( x ) τ ( z ) := θ ( z − / − τ / − x ), we have the following result: Theta functions areanalogous to homogenous polynomials on projective space in that all meromorphic functionson an elliptic curve can be described by their ratios. Theorem 4.2.
A function R : E τ → C is meromorphic if and only if it can be written inthe form R ( z ) = (cid:81) di =1 θ τ ( x i ) ( z ) (cid:81) di =1 θ τ ( y i ) ( z ) for { x i } and { y i } any finite sets of d > complex numbers such that (cid:80) i x i − (cid:80) i y i = ∈ Z .Proof. Showing that any ratio of theta functions is well-defined (and thus meromorphic)on E τ if and only if it meets the above criteria can be seen simply by checking when suchratios are invariant under translation by τ ; a straight-forward computation.A simple proof that all meromorphic functions are such ratios can be found in [32]. (cid:3) We are interested in theta functions for the reason that they descend to holomorphicsections of line bundles on E τ and, as we will see below, will give bases for our B-sidehomsets. As we construct our mirror functor in the following sections, we will need tointroduce the following slight generalization of θ . θ [ a, z ]( τ, z ) = (cid:88) m ∈ Z e πi [( m + a ) τ +2( m + a )( z + z )] Notice that θ [ a, τ, z ) = e πi [ a τ +2 az ] θ τ ( z + aτ )so properties (i) and (ii) still hold. In particular θ [ a, τ, z + τ ) = e − πi [ τ +2 z ] θ [ a, τ, z ) . Also later we will need to use that θ [ a + b, τ, z ) = e πi [ a τ +2 az ] θ [ b, τ z ]( τ, z + aτ ) . Thinking of theta functions as line bundle morphisms, the following addition formula (II.6.4 of [34]) will all allow us to compute compositions on the B-side.
Proposition 4.3.
Let a, b ∈ Q , n , n ∈ Z ≥ , k = n + n , and c j = jn + a + b . θ [ a/n , n τ, z ) · θ [ b/n ,
0] ( n τ, z )= (cid:88) j ∈ Z /k Z θ (cid:104) c j k , (cid:105) ( kτ, z + z ) · θ (cid:20) n c j − kbn n k , (cid:21) ( n n kτ, n z − n z )For example, if n = n = 1, a = b = 0 and z = z = z − x , this gives us(4.1) θ τ ( z ) · θ τ ( z + x ) = θ τ ( x ) θ τ (2 z + x ) + θ [1 / , τ, x ) θ [1 / , τ, z + x ) N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 17
The Category of Holomorphic Line Bundles on Elliptic curves.
Letting Pic d ( C ) denote the space of degree d line bundles on a curve C of genus g and L d ∈ Pic d ( C ), we have an isomorphism J ( C ) := Pic ( C ) ∼ −→ Pic d ( C ) given by L (cid:55)→ L⊗ O C L d .By the Abel-Jacobi theorem J ( C ), usually called the Jacobian variety of C , is isomorphicto a complex torus of dimension g . In the case of an elliptic curve E τ , we then have that E τ ∼ = J ( E τ ). To see this we can use the line bundle-divisor correspondence and the aboveisomorphism J ( E τ ) ∼ = Pic ( E τ ). The degree one line bundles L x and L y associated withdivisors x ∈ E τ and y ∈ E τ are equivalent if and only if x = y . The homomorphism L x : E τ → Pic ( E τ ) is then 1-1. It is also surjective (and thus prove this case of theAbel-Jacobi theorem) we need only to observe that, combining the Riemann-Roch theoremand Serre duality, deg( L ) = h ( L ) and so, if deg( L ) = 1, then L has a global holomorphicsection vanishing only at one point.Note as we continue to discuss holomorphic line bundles, we will generally refer to them assimply line bundles and their global holomorphic sections as simply, sections.By the construction of this isomorphism, fixing any degree one line bundle, L , we canparameterize all degree n line bundles by ( t ∗ x L ) ⊗ L n − , where t x is the automorphism of E τ given by translation by any particular x ∈ E τ and L k := L ⊗ k is the k th tensor productof the line bundle with itself.For our particular task of building a mirror functor we will pick a special L . In particular,the one which holds θ τ as a section.For convenience now we will use our reparameterization E q := C ∗ /u ∼ qu .Any holomorphic function ϕ : C ∗ → C ∗ gives rise to a line bundle L q ( ϕ ) := C ∗ × C / ( u, v ) ∼ ( uq, ϕ ( u ) v ) . In fact, all line bundles on E q can be described in this way and L ( ϕ ) ∼ = L ( ϕ (cid:48) ) if and onlyif there exists some holomorphic B : C ∗ → GL ( C ) such that ϕ (cid:48) ( u ) = B ( qu ) ϕ ( u ) B − ( u ).To see this, note that, as all line bundles on C ∗ are trivial, there exists a global frame onthe pullback π ∗ L ( ϕ ). Given such a frame, ϕ can be thought of as the induced fiber map( π ∗ L ( ϕ )) u → ( π ∗ L ( ϕ )) qu . If L ( ϕ ) ∼ = L ( ϕ (cid:48) ) then there must exist some ˆ B : L ( ϕ ) ∼ −→L ( ϕ (cid:48) )such that the following diagram commutes.( π ∗ L ( ϕ )) uπ ∗ ˆ B ( u ) (cid:15) (cid:15) ϕ ( u ) (cid:47) (cid:47) ( π ∗ L ( ϕ )) quπ ∗ ˆ B ( qu ) (cid:15) (cid:15) ( π ∗ L ( ϕ (cid:48) )) u ϕ (cid:48) ( u ) (cid:47) (cid:47) ( π ∗ L ( ϕ (cid:48) )) qu The fact that any such B descends to and isomorphism of line bundles on E q can beshown by simply verifying that the line bundle L ( ϕ ) is well-defined for any ϕ .Below we will often make use of the fact that L ( ϕ ) ⊗ L ( ϕ ) = L ( ϕ ϕ ). Recalling thatthe tensor product of two line bundles gives a line bundle whose transition functions aregiven by the product of the respective transition functions of those bundles, this fact is clear. L x ∼ = L y ⇐⇒ x = y is true on any curve C (cid:29) P . We can think of any meromorphic function f on C as holomorphic map f : C → P . If ord ( f ) = x − y , then this holomorphic map is degree one and thus anisomorphism C ∼ −→ P . Consider in particular L ϕ := L ( ϕ ), where ϕ ( u ) = q − / u − . Notice that, by 4.1 above, θ q ( z ) descends to a holomorphic section of L . This is our special choice of degree one linebundle to parameterize all others.Recall that, for positive degree line bundles on the elliptic curve, h ( L ) = deg( L ) > n >
0, we similarly have that a basis of H ( L n ) is given by θ [ j/n, nτ, nz ) for j ∈ Z /n Z .From the above parameterization of the Picard group, it is also clear that we have t ∗ x L ϕ = L ( t ∗ x ϕ ). Together, these facts allow us to completely describe the additive structure of thecategory of line bundles on an elliptic curve.In particular, let L = ( t ∗ x L ϕ ) ⊗ L n − ϕ and L = ( t ∗ y L ϕ ) ⊗ L m − ϕ be any two line bundlesover E τ .Then Hom( L , L ) = H ( L ∗ ⊗ L ) = H (( t ∗ x L ∗ ϕ ⊗ t ∗ y L ϕ ⊗ L m − nϕ )Using t ∗ δ ϕ k = e − πiδk ϕ k , we can rewrite this bundle as t ∗ x L ∗ ϕ ⊗ t ∗ y L ϕ ⊗ L m − nϕ = L ( t ∗ x ϕ − · t ∗ y ϕ · ϕ m − n ) = L ( e πi ( x − y ) ϕ m − n ) = t ∗ δ L kϕ for k := m − n and δ := y − xk and where the last equality holds only if k (cid:54) = 0.If L and L are of the same degree (i.e. k = 0), then any morphism between themcorresponds to a global section the degree zero line bundle L ∗ ⊗ L . Non-zero sections ofdegree zero line bundles are non-vanishing and thus correspond to an isomorphism. Thisgives us a contradiction unless x = y . Recalling that only positive degree bundles admitsections, the Hom( L , L ) is non-zero if and only if m > n or both n = m and x = y .To summarize, the only nontrivial case we have is when k := m − n > δ := y − xk .In which case we have,Hom( L , L ) = H ( t ∗ δ L kϕ ) = span C { θ [ j/k, kδ ]( kτ, kz ) } j ∈ Z /k Z Remark 4.4.
One can check in fact that t ∗ x is functorial and thus defines a family (pa-rameterized by the torus) of functorial auto-equivalences of the category of line bundles on E τ .Let us introduce the less cumbersome notation f ( k ) j ( z ) := θ [ j/k, kτ, kz ).5. The A-side
B-fields and the K¨ahler Moduli Space.
Let (
X, J ) be a complex manifold and define K ( X, J ) := { [ ω ] ∈ H ( X, R ) | ω is K¨ahler } t ∗ x takes homsets to homsets in the obvious way and commutes with the composition(product) of theta functions. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 19
This is usually called the
K¨ahler cone of X . Mirror symmetry suggests that we shouldconsider the K¨ahler moduli space, of ( X, J ), as (the complex manifold): M K¨ahler ( X, J ) := ( H ( X, R ) ⊕ i K ( X, J )) /H ( X, Z )A closed 2-form ω C = B + iω representing an element in this space is called a complexifiedK¨ahler form and this tacked on closed 2-form B is called a B-field . As H ( E τ , R ) = R , our complexified K¨ahler form is determined by a complex number ρ = B + iA , with A >
0. We will from now on denote the (real) 2-torus equipped with thiscomplexified K¨ahler form by E ρ . Remark 5.1.
Homological mirror symmetry is a relationship only conjectured to existbetween a K¨ahler manifold and its mirror dual. That said, the Fukaya category will onlydepend on the symplectic structure and the derived category of coherent sheaves will onlydepend on the complex structure (of the mirror dual).5.2.
The Fukaya Category.
It is not clear exactly what the objects of the Fukaya category should be. Fukaya definedthe category [13, 15] over (a countable set of) closed Lagrangian submanifolds equippedwith line bundles of curvature B | L . Kontsevich, inspired by string theoretic D-branes, sug-gested taking these objects as closed special Lagrangian submanifolds equipped with flatcomplex line bundles with unitary connections. Also, it is, in general, necessary to addsome additional structure for purposes of grading and further require that our Lagrangiansubmanifolds are relatively spin. This latter condition is automatically satisfied in the one-dimensional case - the (relative) spin structure is a Z choice here ( H ( L ; Z ) in general)and can be suppressed.In the case of elliptic curves, each Hamiltonian isotopy class of closed Lagrangian sub-manifolds has a unique special Lagrangian representative. This can be shown using theRicci flow and in fact, using our mirror functor to suggest a definition of “stable” specialLagrangian objects, this given a symplectic analogue of Atiyah’s classification of vectorbundles on an elliptic curve [24, section 38.4]. For a general Calabi-Yau manifold, it wasshown in [41] that this uniqueness (but not necessarily existence) is always true when theFukaya category is unobstructed. As shown in section B, the closed special Lagrangiansubmanifolds of E τ are exactly the geodesics (i.e. the projections, through C → E τ , ofrationally sloped lines). Following the middle-ground taken by [37, 31], we will equip ourLagrangian submanifolds with a local systems whose monodromy has eigenvalues of unitmodulus (i.e. complex vector bundles equipped with connections whose holonomy group isgenerated by fiber automorphisms with eigenvalues of unit modulus). The Jordan blockswill be related to non-stable vector bundles on the B-side. B andnot from reference to Type IIB string theory (the B-side of mirror symmetry). The mathematical meaning of B-fieldshas been studied by Hitchin and others in the newly developing field of “generalized geometry.” The Objects.
In specific, here we define the objects of
F uk ( E ρ ) to be the graded, oriented, closedgeodesics (of E ρ ) equipped with a flat complex vector bundle of monodromy with uniteigenvalues (in the sense discussed in remark 2.10). All Lagrangian submanifolds of a 2-surface are of course homeomorphic to S and, as shown in section 2, all complex vectorbundles on S are trivial. Moreover, we know from corollary 2.9, that any flat connection ona complex vector bundle over S is determined (up to gauge equivalence) by the conjugacyclass (in GL k ( C )) of its monodromy operator M .We will give a grading to our special Lagrangian submanifolds, L , determined by a choice α ∈ R such that z ( t ) = z + te πiα pararameterizes a lift of L to C (for some appropriate z ). α also determines an orientation on L given by the direction of e πiβ for the unique β ∈ ( − / , /
2] such that α − β ∈ Z . So, following the notation in [31], we will use tuples L = ( L, α, M ) to denote the objectsof
F uk ( E ρ ). And we will often use M p to refer to the stock of our local system (i.e. fiberof our vector bundle) over a point p ∈ L . We have a shift functor, which mirrors that in D b Coh( E ρ ), given by ( L, α, M )[1] := (
L, α + 1 , M ))5.4.
The Morphisms.
Suppose that L and L (cid:48) are two objects such that L (cid:54) = L (cid:48) .Hom( L , L (cid:48) ) := (cid:77) p ∈ L ∩ L (cid:48) Hom( M p , M (cid:48) p )Similarly if L = L (cid:48) , we can defineHom( L , L (cid:48) ) := Hom( M, M (cid:48) ) , where Hom( M, M (cid:48) ) is the space of vector bundle morphisms between the associated complexvector bundles (modulo isomorphisms of L and L (cid:48) ).We could of course write these two cases together asHom( L , L (cid:48) ) := Hom( M | L ∩ L (cid:48) , M | L ∩ L (cid:48) )A Z -grading is put on morphisms L → L (cid:48) by the Maslov-Viterbo index, which in thiscase is given by µ ( L , L (cid:48) ) = ceil( α − α (cid:48) ) = − floor( α (cid:48) − α ) ∈ Z N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 21
The A ∞ -structure. The Fukaya category is not a true category in the sense that the composition of mor-phisms is not associative. It does, however, admit an A ∞ -structure.Let R be some commutative ring (for mirror symmetry, we typically take C or Q ).An A ∞ category is given by a set of objects Ob , a graded free R -module Hom ( c , c ) foreach c , c ∈ Ob , and a family of degree 2 − k operations m k : Hom ( c , c ) ⊗ ... ⊗ Hom ( c k − , c k ) → Hom ( c , c k )which satisfy the “ A ∞ associativity relations” for k ≥ n (cid:88) r =1 n − r +1 (cid:88) s =1 ( − (cid:15) m n − r +1 ( a ⊗ ... ⊗ a s − ⊗ m r ( a s ⊗ ... ⊗ a s + r − ) ⊗ a s + r ⊗ ... ⊗ a n ) = 0for all n ≥
1, where (cid:15) = ( r + 1) s + r ( n + (cid:80) s − j =1 deg( a j )).From n = 1, we have m = 0 and giving us a differential.From n = 2, we have m ( m ) = m ( ⊗ m ) + m ( m ⊗ ), which tells us that m defines amultiplication operator satisfying the Leibniz rule of m .From n = 3, we have that m ( m ⊗ ) − m ( ⊗ m ) = m ( m ) + m ( m ⊗ ⊗ ) + m ( ⊗ m ⊗ ) + m ( ⊗ ⊗ m ), which tells us that m is associative “up to a homotopy” m (inthe sense described for chain maps in appendix A.1). In particular this third condition says m is associative at the level of cohomologies. Remark 5.2.
In our case, the Fukaya category on an elliptic curve, we will have that m = 0and thus m will be associative. Generally speaking the Fukaya category is not associative.Typically when constructing the Fukaya category we also need to use a curved (also called obstructed ) A ∞ structure. This means we add a map m (and start the sum with r = 0)obstructing our differential category structure. For example, the first two relations wouldthen be m ( m ) = 0 and m + m ( m ⊗ ) + m ( ⊗ m ) = 0. This obstruction, m , is mirrorto the Landau-Ginzberg superpotential on the B-side [2].The A ∞ -structure of the Fukaya category is given by summing overpseudo-holomorphic disks bounded by Lagrangian submanifolds. To be specific, we willneed to define a particular collection of k + 1-pointed (pseudo-)holomorphic disks. Whilewe do this we will use an index j ∈ Z / Z k +1 . Remark 5.3.
The Fukaya category generally involves pseudo-holomorphic disks, but asevery almost complex structure on a surface is integrable, we can drop the pseudo in ourone dimensional case.Let L j be a collection of k objects and fix a point p j ∈ L j ∩ L j +1 . As we are using theindex j ∈ Z / Z k +1 , p k ∈ L k ∩ L . Let D := { z ∈ C | z ≤ } and let ( D, S k +1 ) denote a diskwith k + 1 marked points S k = (cid:8) e iθ j (cid:9) ⊂ ∂D such that 0 = θ < θ < ... < θ k < π . CM k +1 ( X ; L , ..., L k ; p , ..., p k ) := (cid:40) ( φ, S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ : D → X is pseudo-holomorphic, φ ( e iθ j ) = p j , φ ( e it ) ∈ L j ∀ t ∈ ( θ j − , θ j ) and ∀ j (cid:41) / ∼ Where ( φ, S k +1 ) ∼ ( φ (cid:48) , S (cid:48) k +1 ) if there exists a biholomorphic map f : D → D such that φ (cid:48) ◦ f = φ and f ( e iθ j ) = e iθ (cid:48) j ∀ j . Remark 5.4.
The ordering of θ ’s ensures that as we traverse the boundary of the disk,we hit the marked points in order. Parallel transport along this boundary will define our A ∞ “composition maps”.Let u (cid:96) ∈ Hom( L (cid:96) − , L (cid:96) ) for (cid:96) = 1 , .., k be a collection of k morphisms and assume theinvolved k + 1 Lagrangian submanifolds are distinct. We will often refer to this as thetransversal case .For notational convenience, we will let morphisms be denoted by sums of the form (cid:80) µ t µ · p µ where each p µ is an intersection point and each t µ is a morphism between fibersover p µ . m k ( u ⊗ ... ⊗ u k ) = (cid:88) p ∈ L k ∩ L C ( u , ..., u k ; p ) · p where the fiber morphisms are defined by C ( u , ..., u k ; p ) = (cid:88) [ φ ] e πi (cid:82) φ ∗ ω C h ∂φ ( u ⊗ ... ⊗ u k )This second sum is over equivalence classes [ φ ] in the set ∈ CM k +1 ( E ρ ; L , ..., L k ; p , ..., p k − , q ) and h ∂φ ( u ⊗ ... ⊗ u k ) := P γ k ◦ u k ◦ ... ◦ u ◦ P γ , where P γ j : ( M j ) p j − → ( M j ) p j is the parallel transport operator (induced by the connec-tion of L j ) over the curve γ j ( t ) = φ ( e it ), t ∈ [ θ j − , θ j ].We will not check that these satisfy the A ∞ relations here. We will give some discussionof the non-transversal case, but only after descending to the zeroth cohomology. For a morecomplete discussion of this category in the case of elliptic curves (and more generally forabelian varieties) see [14]. Remark 5.5.
The construction of the Fukaya category on a general manifold has manytechnical obstructions. The relevant obstruction theory and complete definition can befound in [15]. We mentioned previously the addition of a (relative) spin structure to ourLagrangians, which is to solve a problem of orientability of the moduli space of disks.Another difficulty in higher dimensions is the existence of pseudo-holomorphic bubbling.This is typically solved by replacing the complex coefficients of our homsets with a universalNovikov ring (where the above m k sums converge). There is an alternative approach givenby cluster algebras in [8].The following basic property of topological covering spaces will help us count holomor-phic discs. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 23
Lemma 5.6 (The Lifting Criterion) . Let ˜ X p −→ X be any covering space and ϕ : Z → X amap to its base from a path-connected and locally path-connected space Z . Fix a base point z ∈ Z and a corresponding fiber point ˜ x ∈ p − ( ϕ ( z )) .There exists a lift ˜ ϕ : Z → ˜ X of ϕ such that p ◦ ˜ ϕ ( z ) = ϕ ( z ) if and only if the inducedhomomorphisms ϕ (cid:93) : π ( Z, z ) → π ( X, ϕ ( z )) and p (cid:93) : π ( ˜ X, ˜ x ) → π ( X, ϕ ( z )) are such thatthe image of the former is contained in the image of the later. I.e. ϕ (cid:93) ( π ( Z, z )) ⊂ p (cid:93) ( π ( ˜ X, ˜ x )) Moreover, if such a lift ˜ ϕ exists, it is unique.Proof. This result can be easily gotten from the homotopy lifting property. A detailed proofcan be found in [20]. (cid:3)
Remark 5.7.
Notice that if Z is simply connected (e.g. a disc), this condition is triviallysatisfied. Thus we have that, any mapping of a k -pointed disk into E ρ (like those usedto define m k ) lifts to a k -gon in C with sides lying on lines lifted from k geodesics. As π : C → E τρ is a holomorphic map, we know, by the Riemann mapping theorem, that each k -pointed disc in C represents such a map. This is well-defined as the lifting operation isequivariant with respect to automorphisms of the disk (i.e. (cid:93) φ ◦ f = ˜ φ ◦ f ). This gives usan easy way to count these holomorphic disks and also tells us that m = 0. As m acts asthe differential operator in this category, this means our cohomology complex is identicalto our complex of homsets graded by the Maslov-Viterbo index 5.4. Remark 5.8.
From corollary 2.9, we can assume (as our Lagrangians are just circles) thatthe above parallel transport operators P γ are given by P γ = exp( M (cid:96) ), where M is themonodromy operator and (cid:96) is the length of γ divided by the length of the Lagrangian (e.g.if γ covers the Lagrangian, this is the winding number).6. The Simplest Example
As we will see in the following sections, the heart of homological mirror symmetry forelliptic curves is a functor between holomorphic vector bundles (and skyscraper sheaves)over E τ and complex local systems over non-vertical (and, respectively, vertical) geodesics of E τ . Let us take a look at how holomorphic line bundles will be sent to lines of integer slope.More specifically, our functor takes gives the following correspondence between objects.A-side Objects B-side Objectsrank 1 local systems of holomorphic line bundlesslope d , y -intercept y , and ( t ∗− y τ + β L ϕ ) ⊗ L d − ϕ connection ∇ = d − πiβdx Except in the horizontal case, we could write the corresponding holomorphic line bundlein terms of the x -intercept, x , as ( t ∗ dx τ + β L ϕ ) ⊗ L d − ϕ = t ∗ x τ + βd L dϕ .So what about morphisms? Two geodesics of respective slopes n and n with respective x -intercepts x and x have intersections at points e k = (cid:18) n x − n x + kn − n , n n − n [ n ( x − x ) + k ] (cid:19) for k ∈ Z / ( n − n ) Z We then want to send these points to linear combinations of morphisms t ∗ δ τ + β f ( n − n ) k , for δ := n x − n x n − n and β := β − β n − n . Our functor will be given by t ∗ δ τ + β f ( n − n ) k (cid:55)→ e πiτ ( n − n ) δ · e k Remark 6.1.
Note that we can not simply think of this as sending theta functions to pointsor vise versa. We will give below an example where multiple theta functions are sent to thesame point (but each time that point is thought of as living in a different homset). Thatsaid, t ∗ δτ + β f ( n ) k is always going to be sent to a point with x -coordinate δ + k/n and coeffi-cient e − πinτδ . If we imagine translating one Lagrangian in the x -direction while keepingthe other fixed, we see this translation must correspond to a change in δ of the same amount.We only have one direction to translate in on the A-side, but we can also twist ourline bundle, which we see corresponds to changing β by the proportional amount. Themonodromy (i.e. twisting) can be easily visualized by thinking of L × C ∼ = S × D as asolid donut containing a curve (not intersecting the zero curve) representing the paralleltransport of the unit once around the circle. We can picture the rank k situation byimagining k (ordered) curves on each of k solid donuts, but it is difficult to visualize thenondegeneracy for k >> Let τ = iA ∈ i R and consider three Lagrangian submanifolds L , L , L , whose lifts to R can be chosen to all travel through the origin, and have slopes 0 , , and 2 respectively. Fornow lets assume the trivial connection ∇ = d . The grading (i.e. choice of α ) will not yet beimportant, and, for now, we are assuming the trivial connection, so we do not need to worryabout anything but the submanifold itself. The utility of the grading is mostly in finding acategorical equivalence (we need a shift functor on the A-side and the grading will play thatrole). In the simple example we are discussing here, we will only need the grading for theorientation it endows on our Lagrangian submanifolds. For most of this simple example,we will assume trivial connections, which will allow us to suppress the choice of grading alltogether. Figure 6.1.
The bold lines mark our three Lagrangian submanifolds. Dotted linessignify repetitions.
On the B-side, these objects will correspond to line bundles L = O , L = L ϕ , and L = L ϕ . Note L ϕ := L ( ϕ ) is our special choice of the degree 1 line bundle defined N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 25 above.We then want our functor to send C | L i ∩ L j | = Hom( L i , L j ) → Hom( L i , L j ) = H ( L j ⊗ L ∗ i )in a way that is compatible with composition.Observe thatHom( L , L ) ∼ = H ( L ϕ ) = span C ( { θ τ ( z ) } )Hom( L , L ) ∼ = H ( L ϕ ) = span C ( { θ τ ( z ) } )Hom( L , L ) ∼ = H ( L ϕ ) = span C ( { θ [0 , τ, z ) , θ [1 / , τ, z ) } )Composition of morphisms is given by fiberwise multiplication. So the above additionformula, 4.3, determines all compositions on the A-side of this simple example.Let us take a look at the A-side and find our m composition operator. From figure 6,we can see L ∩ L = { e } L ∩ L = { e } L ∩ L = { e , e } Recall our lifting lemma (and remark 5.7) above. Fixing the lift of e to be the originin C , our triangles are determined by the winding number (including orientation ) of theboundary segment mapped to L .Notice that when immersion γ (cid:35) L has an even winding number, all corners of thetriangle descend to the same point in E τ . In the odd case, e = π (1 / ofthese triangles respectively are φ ∗ ω C = τ n = iAn and φ ∗ ω C = τ n = iA ( n + 1 / , where n is the winding number of γ (cid:35) L . m ( e , e ) = C ( e , e , e ) · e + C ( e , e , e ) · e Where C ( e , e , e ) = (cid:88) n ∈ Z e − πAn C ( e , e , e ) = (cid:88) n ∈ Z e − πA ( n +1 / Notice that C ( e , e , e ) = θ [0 , iA,
0) = θ τ (0) and C ( e , e , e ) = θ [1 / , τ, m ( e ⊗ e ) = θ τ (0) · θ τ (2 z ) + θ [1 / , τ, · θ [1 / , τ, z ) ∈ H ( L ϕ ) =Hom( L , L ).In fact, by the addition formula above, m ( e ⊗ e ) = [ θ τ ( z )] .This suggests our functor might sendHom( L , L ) → Hom( L , L ) e (cid:55)→ θ τ ( z ) L by its grading (i.e. choice of α ).26In this case with trivial B-field, φ ∗ ω C gives the area of the lifted disk, but in general this quantity will be complexand is often referred to as the energy of φ . Hom( L , L ) → Hom( L , L ) e (cid:55)→ θ τ ( z )Hom( L , L ) → Hom( L , L ) (cid:26) e e (cid:55)→ θ [0 , τ, z ) (cid:55)→ θ [1 / , τ, z )Let us now complicate our simple example a bit and see what happens on the A-sidewhen we replace L with another geodesic of slope 2. Let L be the geodesic in E τ descend-ing from the line of slope 2 with x -intercept given by x ∈ (0 , / Let L (cid:48) denote thecorresponding local system with trivial connection. Figure 6.2.
The shifted case.
Now we have m ( e ⊗ e (cid:48) ) = C ( e , e (cid:48) , p ) · p + C ( e , e (cid:48) , p ) · p Where C ( e , e (cid:48) , p ) = (cid:88) n ∈ Z e − πA ( n + x ) = θ [ x , τ, C ( e , e (cid:48) , p ) = (cid:88) n ∈ Z e − πA ( n +1 / − x ) = (cid:88) n ∈ Z e − πA ( n + x +1 / = θ [1 / x , τ, θ [1 / − x , τ,
0) = θ [1 / x , τ, L (cid:48) should be t ∗ x τ L ϕ . So in this situationwe have Hom( L , L ) = span C { θ τ ( z ) } Hom( L , L (cid:48) ) = H ( t ∗ x τ L ϕ ) = span C { t ∗ x τ θ τ ( z ) } x ∈ (0 , /
2) without any loss of generality.
N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 27
Thus our compositions L → L → L (cid:48) are determined by the product θ τ ( z ) · θ τ ( z + 2 x τ ) and should be members of the spaceHom( L , L (cid:48) ) = H ( t ∗ x τ L ϕ ) = span C { θ [ j/ , x τ ](2 τ, z ) } j ∈ Z /j Z From the above addition formula for theta functions, we have θ τ ( z ) · θ τ ( z + 2 x τ )= θ τ (2 x τ ) θ τ (2 z + 2 x τ ) + θ [1 / , τ, x τ ) θ [1 / , τ, z + 2 x τ )= e − πix τ [ θ [ x , τ, θ [0 , x τ ](2 τ, z )+ θ [ x + 1 / , τ, θ [1 / , x τ ](2 τ, z )]This tells us that, our functor preserves composition by sendingHom( L , L ) → Hom( L , L ) (cid:26) p p (cid:55)→ θ [0 , τ, z ) (cid:55)→ θ [1 / , τ, z )Before moving on from this simple example, let us see what happens when we give L anontrivial connection. Let ∇ = d + 2 πiβdt for t some coordinate on L with t + 1 t .Then C ( e , e , e ) = (cid:88) n ∈ Z e − πAn +2 πinβ = θ [ x , β ](2 τ, C ( e , e , e ) = (cid:88) n ∈ Z e − πA ( n +1 / +2 πi ( n +1 / β = θ [ x + 1 / , β ](2 τ, x = 2 x τ + β .It will be left as an exercise to the reader to check that composition is again preserved inthis case. 7. The B-side
The Derived Category of Coherent Sheaves.
Let A be some abelian category and consider, CA the category of chain complexes in A , where objects are chain complexes (of objects in A ) and morphism are chain maps. The homotopy category of A , denoted HA , has the same objects as A , but we consider themorphisms up to homotopy equivalence. The derived category of A , DA , is the localizationof HA over quasi-isomorphisms (i.e. we formally add inverses to quasi-isomorphisms). Inmirror symmetry we always talk about the bounded derived category of coherent sheaveson some variety X , D b Coh( X ) (or often, as we will do here, simply denoted D b ( X )). Thisis the full subcategory of DA given by chain complexes in Coh X ( A ) of finite length. Theseconcepts are explained in more detail in section A.2, but in the case we are interested inhere, D b ( E τ ), little understanding of this general topic is needed. In this section we willcollect some standard facts about D b ( X ) and coherent sheaves on X , which we will use togain a simple description of D b ( E τ ). All results in this section are either standard or provenin [37, 31]. The classic example of a coherent sheaf is the space of local sections of a vector bun-dle. This correspondence gives a natural equivalence between locally free coherent sheavesand vector bundles over manifolds. With this in mind, we will often, in a slight abuse oflanguage, call a (locally free) coherent sheaf a vector bundle or visa-versa. To avoid unnec-essary (in our simple case of elliptic curves) algebraic geometry we will offer the followingtheorem in lieu of the definition of a coherent sheaf . Theorem 7.1.
All indecomposable coherent sheaves over a Riemann surface are eitheran indecomposable vector bundle or a torsion sheaf supported at one point.Proof. See [37] or [19, Ex. III.6.11]. (cid:3)
Remark 7.2.
A torsion sheaf, F , on E τ supported at a single point, ζ ∈ E τ , is determinedby a finite dimensional (complex) vector space V := F ζ and a nilpotent endomorphism N ∈ End ( V ). Following [37, 31], we will denote such an sheaf by S τ ( ζ , V, N ). In otherwords, S τ ( ζ , V, N ) := O rζ ⊗ V / < ζ − ζ − πi N > where O rζ is defined by the exact sequence O ( ζ − ζ ) r −−−−−→ O → O rζ . S τ ( ζ , V, N ) is indecomposable if and only if N has a one dimensional kernel (as thisimplies N r = 0 and N r − (cid:54) = 0 for r = dim( V )).For a object F in D b ( C ), let F [ − n ] denote the chain complex composed entirely of zeroobjects except for its n th degree term, which is F . Theorem 7.3 ([31]) . Let C be a compact Riemann surface. Finite direct sums of objects F [ n ] , where F is an indecomposable coherent sheaf on C , form a full subcategory of D b ( C ) which is equivalent to D b ( C ) . We can state Serre duality on the B-side as follows (see [31] for a proof),
Lemma 7.4.
Let F and G be to coherent sheaves on E τ , then there is a functorial isomor-phism Hom( F , G ) ∼ = Ext ( G , F ) ∗ . For any
A, B objects of any abelian category we have that Hom DA ( A [ m ] , A [ n ]) ∼ = Hom DA ( A, B [ m − n ]) ∼ = Ext m − n ( A, B ) A • such that A n = 0 for some n >>
0. Also, when defining the D b Coh( X ), we technically use cochain complexes. A sheaf (or vector bundle) X is indecomposable if X ∼ = X ⊕ X ⇒ X ∼ = 0 or X ∼ = 0.30We will often abuse notation by denote F [0] = F . Also, if F is already a chain in some category of complexes, wewill use F [ − n ] to denote the corresponding shifted chain.31This is often taken as a definition, see [28] or [18, section I.6] for an explanation. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 29
Letting our abelian category be that of coherent sheaves on E τ and using the above lemmathis gives us functorial isomorphismsHom D b ( E τ ) ( A, B [1]) = Ext ( A, B ) = Hom(
B, A ) ∗ Hom D b ( E τ ) ( A, B ) = Ext ( A, B ) := Hom(
A, B ) Lemma 7.5.
For
A, B coherent sheaves on a Riemann surface,
Ext k ( A, B ) = 0 unless k ∈ , .Proof. In general we can say that for a complex n -dimensional manifold the Ext k functorvanishes for k > n . See [19, Ex. III.6.5]. (cid:3) We will discuss a mirror to Serre duality in section 8.7.2.
The Classification of Holomorphic Vector Bundles on E τ . Let V be an finite dimensional (complex) vector space and consider the holomorphicvector bundle on E q given by F q ( V, A ) = C ∗ × V / ( u, v ) ∼ ( uq, A · v )for any A ∈ GL ( V ).As all homomorphic vector bundles on C ∗ are trivial, the related argument in section 4.2still holds that we can describe all line bundles in this way (uniquely up to B : C ∗ → GL( V )such that A ( u ) = B ( qu ) A ( u ) B ( u ) − ).If A = exp( N ) for N a nilpotent endomorphism with a one dimensional kernel, then F q ( V, exp( N )) is indecomposable. Moreover, we have the below classification (theorem 7.6)by Atiyah [1].Consider r -fold covering π r : E q r → E q . π r defines a functor through its pullback andpushforward which will commute with our mirror functor (we will discuss this functor andits mirror in more detail later (see section 8.1). Theorem 7.6.
Every indecomposable holomorphic vector bundle on E τ is of the form π r ∗ ( L q r ( ϕ ) ⊗ F q r ( V, exp( N ))) for some ϕ and some nilpotent N with a one dimensionalkernel. In the next section we will formally add biproducts on the A-side and so when definingour functor (and equivalence of additive categories), it will be sufficient to define it onthese indecomposable bundles. Moreover, we will have that this equivalence commuteswith π ∗ and thus only have to consider objects of the form L ( ϕ ) ⊗ F ( V, exp( N )). Once thefunctorality of π ∗ is explicitly established, the following proposition tells us our homsets canbe again be dealt with using bases of theta functions (with constant vector coefficients). Lemma 7.7.
Let ϕ = ( t ∗ x ϕ ) · ϕ n − for some n > . Then for any nilpotent N ∈ End( V ) ,there is a canonical isomorphism Ψ ϕ,N := H ( L ( ϕ )) ⊗ V ∼ −→ H ( L ( ϕ ) ⊗ F ( V, exp( N ))(7.1) N ) is always invertible as det( e N ) = e Tr(N) . f ⊗ v (cid:55)→ exp( DN/n ) f · v (7.2) Where D = − u ddu = − πi ddz .Proof. This is simple to check. See [37]. (cid:3)
Consider two vector bundles V i = L ( ϕ i ) ⊗ F ( V i , exp( N i )) with ϕ i = ( t ∗ x i ϕ ) · ϕ n i − and n < n , using lemma 7.1, we then haveHom( V , V ) = H ( L ( ϕ − ϕ ) ⊗ F ( V ∗ ⊗ V , exp( ⊗ N − N ∗ ⊗ ) ∼ = H ( L ( ϕ − ϕ )) ⊗ Hom( V , V )Note: F ( V, A ) ∗ = F ( V ∗ , ( A − ) ∗ ) ∼ = F ( V, A − ).8. The Equivalence
The Fukaya-Kontsevich Category.
The equivalence proven in [37, 31] is not that of A ∞ -categories, but additive categories D b ( E τ ) and F K ( E τ ) := H ( F uk ( E τ )). Let us now explain what we mean by this A-sidecategory.As we showed previously (see remark 5.7), the differential operator on
F uk ( E τ ) is trivial.Thus the cohomology complex is identical to the complex given by homsets as gradedby the Maslov-Viterbo index 5.4. First, following [37], lets descend/restrict to the trivialcohomology, i.e. F uk := H ( F uk ( E τ )), where we have the same objects, but only considermorphisms ( L, α, M ) → ( L (cid:48) , α (cid:48) , M (cid:48) ) such that µ L , L (cid:48) = 0 (i.e. α (cid:48) − α ∈ [0 , L (cid:54) = L (cid:48) ), we have Hom
F uk ( E τ ) ( L , L (cid:48) ) = (cid:40) α (cid:48) − α / ∈ [0 , (cid:76) p ∈ L ∩ L (cid:48) Hom( M p , M (cid:48) p ) if α (cid:48) − α ∈ [0 , L = L (cid:48) , we haveHom F uk ( E τ ) ( L , L (cid:48) ) = α (cid:48) − α / ∈ , H ( L, Hom(
M, M (cid:48) )) = Hom(
M, M (cid:48) ) if α (cid:48) = αH ( L, Hom(
M, M (cid:48) )) if α (cid:48) = α + 1Descending to H ( F uk ) leaves us with a true (i.e. associative) preadditive category. In order to have equivalence with the entire derived category on the B-side, we must alsoconsider the above degree one morphisms (see [31] for further details). m k in the case of k + 1 distinct Lagrangian submanifolds. [37, 36] only deal with thiscase and [31] sidesteps the higher A ∞ operators, m k for k ≥
3, altogether by constructing
F uk directly. A definitionof m k in the general case is given by [14, 15], but to the author’s knowledge, no proof of equivalence between A ∞ -(or triangulated) categories in the general (non-transversal) case has been published. It is likely, however, this proofis thought implicit in the combined works of [36, 14].34We will generally drop the F uk from our homsets when it is clear from context. Same goes for the B-side.35We will ignore the higher A ∞ operators, m k for k ≥
3, as we are not seeking an equivalence of A ∞ -categories inthis weakened case of HMS. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 31
Recall that in
F uk ( E τ ), ( L, α, M )[1] := (
L, α + 1 , M ).A proof of the following “symplectic Serre duality” can again be found in [31].
Lemma 8.1.
For L , L objects in F uk ( E τ ) , there exists a canonical isomorphism Hom( L , L [1]) ∼ = Hom( L , L ) ∗ F uk ( E τ ) is not additive (it is preadditive by construction). If (biproduct) L = L ⊕ L exists for some nonzero objects L k , then we have nonzero projections and embed-dings p k : L → L k and i k : L k → L . This is impossible (in the case of distinct Lagrangiansubmanifolds) as we’ve restricted to just the zero graded morphisms, so Hom( L , L ) andHom( L , L ) can not both be non-trivial. This is a problem as we are looking to showequivalence with the additive category D b ( E τ ). [31] fixed this by inserting these biprod-ucts formally. We call this formally enlarged category the Fukaya-Kontsevich category,
F K ( E τ ) := H ( F uk ( E τ )).Let A be a preadditive category. We define A to be the category whose objects areordered k -tuples ( k ≥
0) of objects in A and whose morphisms are matrices of morphismsin A . Composition is given by matrix multiplication and the zero tuple is the zero object.8.1. The Mirror Isogenies.
Before we construct the functor between D b ( E τ ) and F K ( E τ ), we need to define p r : E rτ → E τ the mirror of π r .Recall that π r is the r -fold covering E rτ π r −→ E τ . The pullback of a vector bundle through π r is given by π ∗ r F q ( V, A ) → F q r ( V, A r )and the pushforward is given by π r ∗ F q r ( V, A ) = F q ( V ⊕ r , π ∗ A )where π ∗ ( A ) ∈ GL ( V ⊕ r ) is sends ( v , ..., v r ) (cid:55)→ ( v , ..., v r , Av ).These functors are defined similarly on indecomposable torsion sheaves S τ ( ζ , V, N ) := O rζ ⊗ V / < ζ − ζ − πi N > The mirror to this functor, p r : E τ → E rτ , is induced by the automorphism ( x, y ) (cid:55)→ ( rx, y ) of the real plane (for any positive integer r ). Let us look at the restriction of p r to aLagrangian submanifold, L . Let L have slope n/m with gcd( n, m ) = 1. As p r fixes the y ,
36A category is called preadditive (or an Ab -category ) if its homsets are abelian groups and the composition ofmorphisms is bilinear. It is called additive if it additionally it has a zero object and contains all finite biproducts. the degree of p r | L is given by d = gcd( n, r ). We then must also have that the preimageof a Lagrangian submanifold (of slope n/ ( rm )) has N = r/d = r/ gcd( n, r ) connected com-ponents. I.e. p − r ( L ) = { L (1) , ..., L ( N ) } (we will use this to define the pullback in a moment).The pushforward p r ∗ : F K ( E τ ) → F K ( E rτ ) is then given by p r ∗ ( L, α, M ) (cid:55)→ ( p r ( L ) , α (cid:48) , p r ∗ M )where p r ∗ : ( v , ..., v d ) (cid:55)→ ( v , ...v d , M v ) and α (cid:48) is the unique appropriate value in the sameinterval [ k − / , k + 1 /
2) that α is contained in (for some k ∈ Z ).Functorality is not hard to see.Hom( L , L (cid:48) ) p r ∗ (cid:15) (cid:15) = (cid:76) ˜ x ∈ L ∩ L (cid:48) Hom( M ˜ x , M (cid:48) ˜ x ) (cid:127) (cid:95) (cid:15) (cid:15) Hom( p r ∗ L , p r ∗ L (cid:48) ) = Z Where Z := (cid:77) x ∈ p r ( L ) ∩ p r ( L (cid:48) ) Hom (cid:77) ˜ x ∈ ( p r | L ) − ( x ) M ˜ x , (cid:77) ˜ x (cid:48) ∈ ( p r | L (cid:48) ) − ( x ) M (cid:48) ˜ x (cid:48) Similarly, the pullback p ∗ r : F K ( E rτ ) → F K ( E τ ) is given by p ∗ r ( L, α, M ) = N (cid:77) k =1 ( L ( k ) , α (cid:48) , ( p ( k ) r ) ∗ M )where p ( k ) r := p r | L ( k ) and ( p ( k ) r ) ∗ M is the local system pulled back in the typical way.A check of the following duality between p ∗ r and p r ∗ is given in [31]. Proposition 8.2.
Let p = t (cid:48) n/m ◦ p r for n/m ∈ Q and t (cid:48) δ ( x, y ) := ( x − δ, y ) . Hom( p ∗ L , L (cid:48) ) ∼ = Hom( L , p ∗ L (cid:48) ) and Hom( p ∗ L , L (cid:48) ) ∼ = Hom( L , p ∗ L (cid:48) ) The same is true on the B-side if we replace p by its mirror, π = t nτ/m ◦ π r . After defining our mirror functor, Φ : D b ( E τ ) → F K ( E τ ), on coherent sheaves A = L ( ϕ ) ⊗ F ( V, exp( N )) as in theorem 7.6, we will be able to extend to all indecomposablebundles by proving Φ( π r ∗ A ) = p r ∗ Φ( A )For morphisms, we will need to know how to sendHom( π r ∗ E , π r ∗ E )) → Hom( p r ∗ Φ( E ) , p r ∗ Φ( E ))Consider the categorical pull back of π r and π r , E r τ × E τ E r τ := { ( z , z ) | π r ( z ) = π r ( z ) } . This is a disjoint union of elliptic curves, in particular, E r τ × E τ E r τ = E rτ × Z d for d = gcd( r , r ) and r = lcm( r , r ). n, m ) = 1, L wraps around E τ n times in the y -direction. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 33
Let ˜ π r i ,ν : E rτ × { ν } → E r i τ denote the restriction of the i th term projection. In otherwords, ˜ π r ,ν (resp. ˜ π r ,ν ) is given by composition of the translation t ντ × ( × t ντ ) withthe covering map ˜ p r i : E rτ → E r i τ (defined by the lattice inclusion < , r i τ > ⊂ < , rτ > ).Using the above properties and the fact that ˜ p r ∗ ◦ ˜ p ∗ r and p ∗ r ◦ p r ∗ are isomorphicfunctors (and defining ˜ π r i ,ν analogously), we have the following commutative diagram.Hom( π r ∗ E , π r ∗ E ) Φ τ (cid:15) (cid:15) ∼ −→ d (cid:77) ν =1 Hom(˜ π ∗ r ,ν E , ˜ π ∗ r ,ν E ) (cid:76) Φ τ (cid:15) (cid:15) Hom(Φ( π r ∗ E ) , Φ( π r ∗ E )) (cid:111) (cid:15) (cid:15) d (cid:77) ν =1 Hom(Φ(˜ π ∗ r ,ν E )) , Φ(˜ π ∗ r ,ν E )) (cid:111) (cid:15) (cid:15) Hom( p r ∗ Φ( E ) , p r ∗ Φ( E )) ∼ −→ d (cid:77) ν =1 Hom(˜ p ∗ r ,ν E , ˜ p ∗ r ,ν E ) Remark 8.3.
The significance of this diagram is that by its definition π r i ,ν ( L ( ϕ ) ⊗ F ( V, exp( N )))is also a bundle of the form L ( ϕ ) ⊗ F ( V, exp( N )) (as in Atiyah’s classification). Thus thediagram makes it sufficient to define Φ on just bundles of this form.8.2. The Equivalence of D b ( E τ ) and F K ( E τ ) . To prove the following equivalence (in the transversal case), [37] constructed the functorΦ τ , which we will describe below. Later, by extending this functor to the non-transversalcase and using the additive category F K ( E τ ), [31] made the following rigorous statement. Theorem 8.4 (Main Theorem [37, 31]) . Φ τ : D b ( E τ ) → F K ( E τ ) is an equivalence ofadditive categories compatible with the shift functors. Recall that chain shifts in
F K ( E τ ) correspond to deck transformations α → α + 1.By Serre duality and additivity, it is sufficient to define Φ on indecomposable coherentsheaves. Moreover, with regards to Φ’s image on objects, Atiyah’s classification theorem(coupled with 8.1) tells us we only need to define Φ on only vector bundles of the form A = L ( ϕ ) ⊗ F ( V, exp( N )) and indecomposable torsion sheaves, S = S ( x, V, N ). We will beable to treat morphisms similarly using the above diagram 8.1 and lemma 7.1. We will notrepeat the proof that composition holds, this can be found in [37, 31]. Mirror Objects. If A = L ( ϕ ) ⊗ F ( V, exp( N )) for N ∈ End( V ) nilpotent with nullity one and ϕ = ( t ∗ aτ + b ϕ ) · ϕ n − , then Φ τ ( A ) = ( L, α, M ), where • L is the submanifold of E τ with lift parameterized by t (cid:55)→ ( a + t, ( n − a + nt ). • α is the unique appropriate (i.e. e iπα ) = n + im √ n ) real number such that − < α < . • M = e − πib exp( N ).So in other words we take this rank k -bundle to a null-graded Lagrangian of slope n (thedegree of L ( ϕ )) and y -intercept at − a ( x -intercept at a/n ) with a rank k local system ofmonodromy e − πib exp( N ).Notice how, as in the simple case, (B-side) translations in the real direction correspondto (A-side) shifts of the Lagrangian submanifold and (B-side) translations in the τ -direction(A-side) shift of the monodromy.We extend Φ to all vector bundles by defining Φ( π r ∗ A ) = p r ∗ Φ rτ ( A ) . Notice that asΦ r τ ( A ) has an integer slope, p r | L defines an isomorphism L ∼ −→ p r ( L ). In particular the onlyaffect of p r ∗ on Φ( A ) (for A the particular case above) is sending Lagrangian L (given byline y = nx + a/n ) to p r ( L ) (given by line y = nx/r + ra/n ).Now for the final case of an indecomposable torsion sheaf. Let S = S ( − aτ − b, V, N ), thenΦ τ ( S ) = ( L, / , e − πib exp( N )). Where L is the given by the vertical line with x -intercept a .8.4. Morphisms of Vector Bundles.
For morphisms, we can again use additivity and compatibility with the shift functors toreduce to the case of definingΦ τ : Hom D b ( E τ ) ( A , A [ n ]) → Hom FK ( E τ ) (Φ τ ( A ) , Φ τ ( A )[ n ])for any indecomposable coherent sheaves A and A . By our construction of the Fukayacategory on the A-side, and the fact that we are on a curve on the B-side, both sides of thismap vanish for n / ∈ { , } . Using the isomorphisms then of Serre duality (lemmas 7.4 and8.1), it is sufficient to only consider the case when n = 0.Let A i = L ( ϕ i ) ⊗ F ( V i , exp( N i )) be as in Atiyah’s classification. Using the isomorphismfrom lemma 7.1, we will define Φ as a map Hom( A , A ) ∼ = H ( L ( ϕ − ϕ )) ⊗ Hom( V , V ) → Hom FK ( E τ ) (Φ( A ) , Φ( A )).Using a basis θ ⊗ f for this B-side space, we define Φ as follows (using the notation forsection 6). Let θ = t ∗ δ τ + β f ( n − n ) k , for δ := n x − n x n − n and β := β − β n − n . We defineΦ(Ψ( θ ⊗ f )) = e πiδ ( n − n ) exp[ δ ( N − N ∗ − πi ( n − n ) β )] · f · e k for e k the corresponding intersection point, which is given by e k = (cid:18) n x − n x + kn − n , n n − n [ n ( x − x ) + k ] (cid:19) for k ∈ Z / ( n − n ) Z N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 35
By remark 8.3, this defines Φ on all vector bundle morphisms.8.5.
Morphisms of Torsion Sheaves.
First the mixed cases. Recall that we can write any indecomposable torsion sheaf on E τ in the form S = S ( − aτ − b, V (cid:48) , N (cid:48) ) as defined in section 7. Let S be such a torsion sheafand A = L ( ϕ ) ⊗ F ( V, exp( N )) a locally free sheaf as above with ϕ = t ∗ ατ + β ϕ · ( ϕ ) n − .Hom( A, S ) = 0. On the symplectic side we know the same is true as Φ( S ) has α = 1 / > α A for α A the grading of Φ( A ). Hom( A, S ) = Hom(
V, V (cid:48) ) = V ∗ ⊗ V . The Lagrangianscorresponding to S and A only intersect at a single point. Thus we can define Φ : V ∗ ⊗ V (cid:48) → V ∗ ⊗ V (cid:48) as the vector space operatorΦ τ = e − πiτ ( na − aα )+2 πi ( aβ + bα − nab ) · exp[( na − α ) V ∗ ⊗ N (cid:48) + αN ∗ ⊗ V ]We extend to these morphisms to morphisms to arbitrary vector bundles by the followingcommutative diagram:Hom( A, π ∗ r S ) Φ rτ (cid:15) (cid:15) ∼ −→ Hom( π r ∗ A, S ) Φ τ (cid:15) (cid:15) Hom(Φ( A ) , Φ( π ∗ r S )) (cid:111) (cid:15) (cid:15) ∼ −→ Hom(Φ( π r ∗ A ) , Φ( S )) (cid:111) (cid:15) (cid:15) Hom(Φ( A ) , p ∗ r Φ( S )) ∼ −→ Hom( p r ∗ Φ( A ) , Φ( S ))Now all that remains is to define morphisms between two indecomposable torsion sheaves.Let S i = S ( − a i τ − b i , V i , N i ). Recall these sheaves each have supports at a single point.To have nonzero morphisms, we need that point to be shared. On the symplectic side wehave two vertical lines with x -intercepts a i and monodromy operators M i = e − πib i exp( N i ).Thus we only have the trivial morphism unless a = a in which case these lines are thesame and Hom(Φ( S ) , Φ( S )) = { f ∈ Hom( V , V ) | f ◦ M = M ◦ f } Also note that if b (cid:54) = b , M and M do not share any eigenvalues (as N is nilpotent ithas no nonzero eigenvalues) and thus can not be conjugates of each other in the space ofendomorphisms.If a τ + b = a τ + b , thenHom( S , S ) = { f ∈ Hom( V , V ) | f ◦ N = N ◦ f } = { f ∈ Hom( V , V ) | f ◦ M = M ◦ f } = Hom(Φ( S ) , Φ( S ))This completes our construction of Φ. Appendix A. Homological Algebra Background
In this appendix we will give a minimalist review of the homological algebra used in thisthesis. More thorough reviews of these subjects are given in [28, 40].A.1.
Chain Complexes.
Before we start our discussion of derived categories let us review some basic terminologyused in homology theory.Let A be an abelian category. Definition A.1. A chain complex in A , ( A • , d • ), is a sequence of objects connected bymorphisms (called boundary operators) d n : A n → A n − such that d n ◦ d n +1 = 0 for all n .Associated to any chain complex is a homology, H n ( A • ) = ker d n − / Im d n . Definition A.2. A chain map between to chain complexes ( A • , d A • ) and ( B • , d B • ) is asequence f • of homomorphisms f n : A n → B n such that d Bn ◦ f n = f n − ◦ d An for all n .Chain maps induce maps between homologies. If the induced maps are isomorphisms,then the chain map is called a quasi-isomorphism . Example:
Continuous maps induce chain maps on singular and smooth maps inducecochain maps on de Rham cochain complexes. Definition A.3.
For semantic convenience let us also recall that a morphism between twograded structures A • and B • is said to be homogeneous of degree k if it maps A n → B n + k for all n . Example:
Chain maps are degree 0 maps that preserve the boundary operator, which itselfis defined as a degree -1 map that squares to zero. Cochain maps are degree 0 maps thatpreserve the coboundary operator, which itself is defined as a degree 1 map that squares tozero.
Definition A.4.
We say two chain maps f • : A • → B • and g • : A • → B • are homotopic if there exists some degree -1 map h • : A • → B • such that f • − g • = d B h • + h • d A (i.e. f n − g n = d n − B h n + h n +1 d nA for all n ). Remark A.5.
If two chain maps are homotopic, then they induce the same maps betweenhomologies.
38A category is called abelian if all its homsets are abelian groups and composition of morphisms gives a bilinearoperator. For example, the category of coherent sheaves on a manifold is abelian.39Actually we only need a continuous map between topological spaces admitting a differential structure as de Rhamcohomology cannot tell the difference between homeomorphic spaces.40We will in general omit this word “homogenous” when speaking of such maps as its implication is clear from thecontext.41This name comes the fact that homotopic maps of topological spaces induce these maps in the case of singularchains.
N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 37
A.2.
The Derived Category.Definition A.6.
Let A be some abelian category and consider, CA the category of chaincomplexes in A , where objects are chain complexes (of objects in A ) and morphism arechain maps. The homotopy category of A , denoted HA , has the same objects as A , but weconsider the morphisms up to homotopy equivalence as defined above.Before defining the derived category of A , we need to discuss quasi-isomorphisms andthe rough concept of “localization”. Localization is, roughly speaking, the process of for-mally adding inverse elements to a subset of morphisms that we are “localizing over”. Forexample, in algebraic geometry, we may, wishing to better understand the local geometryof a point (or subvariety) in some variety, localize our coordinate ring over complement, S , of the maximum (prime) ideal (of the coordinate ring, R ) corresponding to that point(subvariety). For this example S is closed under multiplication, which allows us to definethis localization as the ring of fractions over S , S − R := { rs | s ∈ S and r ∈ R } = R × S/ ,where ( r , s ) ( r , s ) ⇐⇒ ∃ t ∈ S such that t ( r s − r s ) ( t is necessary for transitivity). Definition A.7.
In general we define a quasi-isomorphism to be a morphism that descendsto any isomorphism of homologies. To be more specific (so we can define the derivedcategory below), notice that the homology functor H n : CA → A , which sends a chain(chain map) to its n th object (morphism), descends naturally to a functor HA → A . Wewill say a morphism s ∈ Hom HA ( A, B ) is a quasi-isomorphism if the induced morphism H n s : A → B is invertible for all n .Below we will define the derived category of A to be the localization of HA over thespace of quasi-isomorphisms. First we a few observations about quasi-isomorphisms. Lemma A.8. a) Isomorphisms and compositions of quasi-isomorphisms are quasi-isomorphism. b) Each diagram A (cid:48) s ←− A f −→ B (or A (cid:48) f (cid:48) −→ B (cid:48) s (cid:48) ←− B ) of HA , for s (resp. s (cid:48) ) a quasi-isomorphism, can be embedded into a commutative square: A s (cid:15) (cid:15) f (cid:47) (cid:47) B s (cid:48) (cid:15) (cid:15) A (cid:48) f (cid:48) (cid:47) (cid:47) B (cid:48) c) Let f ∈ Mor ( HA ) . Then there exists a quasi-isomorphism s such that sf = 0 if andonly if there exists a quasi-isomorphism t such that f t = 0 .Proof. A proof of this can be found in section 1.6 of [27]. (cid:3)
Remark A.9.
Notice (a) is true in general, but (b) and (c) hold due to homotopy equiva-lence.Now on to defining the title of this section.
Definition A.10.
The derived category of A , denoted DA , has the same objects HA , butthe morphisms, Hom DA ( A, B ), are given by pairs A f −→ B (cid:48) s ←− B , denoted ( f, s ), of amorphism and quasi-isomorphism under the following equivalence. We say ( f, s ) ∼ ( f (cid:48) , s (cid:48) )if and only if there is a commutative diagram in HA : B (cid:48) (cid:15) (cid:15) A f (cid:62) (cid:62) f (cid:48)(cid:48) (cid:47) (cid:47) f (cid:48) (cid:32) (cid:32) B (cid:48)(cid:48)(cid:48) B s (cid:97) (cid:97) s (cid:48)(cid:48) (cid:111) (cid:111) s (cid:48) (cid:125) (cid:125) B (cid:48)(cid:48) (cid:79) (cid:79) We define composition of morphisms in DA by ( f, s ) ◦ ( g, t ) = ( g (cid:48) f, s (cid:48) t ), where g (cid:48) andquasi-isomorphism, s (cid:48) are defined by the following commutative diagram in HA : C (cid:48)(cid:48) B (cid:48) g (cid:48) (cid:61) (cid:61) C (cid:48) s (cid:48) (cid:97) (cid:97) A f (cid:62) (cid:62) B s (cid:97) (cid:97) g (cid:61) (cid:61) C t (cid:96) (cid:96) Lemma A.11.
Let A be an abelian category. Then, for any X, Y ∈ A and k ∈ Z , there isa canonical isomorphism Ext k A ( X, Y ) ∼ = Hom D b A ( X, Y [ k ]) Proof.
This is true essentially by definition and is often taken as such. An explanation canbe found in [28] or in section I.6 of [18]. (cid:3)
Remark A.12.
The derived category has a triangulated structure, which is often used inmirror symmetry. We will not talk about this here. For a review of the derived category ofan abelian category and its triangulated structure, see [28, 40]. roofs or more appropriately, left fractions . N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 39
Appendix B. Minimal, Calibrated, and Special Lagrangian Submanifolds
Below we will define special Lagrangians and a grading which can be placed on them.First we will give some background by discussing minimal and calibrated submanifolds.B.1.
Minimal Submanifolds.
Roughly speaking, a minimal submanifold is a submanifold with the property that smalldeformations of its embedding do not affect its volume. We will be more specific below,but first let us review some basic notions from Riemannian Geometry. For a more in depthdiscussion on these topics, see [44, 23, 22].Let S and M be n and ( n + k )-dimensional smooth manifolds respectively. An isometricimmersion, S (cid:35) M gives us a smooth splitting of the tangent bundle of M into parts tangentand normal to S , T M | S = T S ⊕ N S . The second fundamental form of S (cid:35) M is a symmetricbilinear N S -valued form on S defined in terms of the Levi-Civita connections on M and S as B ( X, Y ) := ∇ MX Y − ∇ SX Y .The mean curvature of S (cid:35) M is then given by the average of the eigenvalues of the secondfundamental form, H = n Trace( B ). S is said to be a totally geodesic submanifold in M if all geodesics in S are geodesics in M .This is true if and only if B ≡ S is said to be a minimal submanifold in M if H ≡ first variation formula of Riemannian Geometry: Theorem B.1.
Let S be a compact Riemannian manifold and f : S (cid:35) M some isometricimmersion with mean curvature H . Let f t for | t | < (cid:15) , f = f , be a smooth family of im-mersions satisfying f t | ∂S = f | ∂S . Let V = ∂f t ∂t (cid:12)(cid:12)(cid:12) t =0 . Then ddt vol ( f t ( S )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = − (cid:90) S < nH, ∂f t ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 > d vol . Proof.
This is straight-forward application of Stoke’s formula and the fact that ddt det( A ( t )) (cid:12)(cid:12) t =0 = Trace( A (cid:48) (0)). See [44] for a detailed proof. (cid:3) Remark B.2.
The R.H.S. of this equation gives the Euler-Lagrange equation H = 0, whichexplains the above definition of a minimal submanifold. Example:
It’s immediate (from the above theorem or definition) that one-dimensional min-imal submanifolds are geodesics. B is symmetric and thus B ( X, Y ) at a point in S dependsonly on the values of X and Y at that point. B.2.
Calibrated Submanifolds.Definition B.3.
Let M be some Riemannian manifold and vol V denote the induced vol-ume form on any given subspace V ⊂ T x M . A k -form, η , is said to be a calibration on M if it is closed and, for all oriented k -dimensional subspaces V ⊂ T x M , η | V = λ vol V forsome λ ≤
1. We say a submanifold
N (cid:44) → M is calibrated with respect to calibration η (or η -calibrated ) if for all x ∈ N , η | T x N = vol V . Proposition:
Let (
M, g ) be a Riemannnian manifold, η a calibration on M , and N acompact η -calibrated submanifold of M . Then N is volume-minimizing in its homologyclass. Proof.
Let [ N ] = [ N (cid:48) ] ∈ H dim ( N ) ( M ) (cid:90) N vol N = (cid:90) N η = (cid:90) N (cid:48) η ≤ (cid:90) N (cid:48) vol N (cid:48) (cid:3) Remark B.4.
This tells us that the volume of calibrated submanifolds is unchanged bysmall variations, in other words, calibrated submanifolds are minimal submanifolds.
Example:
Any submanifold calibrated by a 1-form is a geodesic.B.3.
Special Lagrangian Submanifolds.
In order to define special Lagrangian submanifolds, we need to make the following obser-vation.Let M be a (complex) n -dimensional Calabi-Yau manifold with Calabi-Yau form Ω andK¨ahler form ω . The volume form induced by the K¨ahler metric is then given by ω n /n ! = n ! (cid:80) dz .Recall that a Lagrangian submanifold is a (real) n -dimensional submanifold L (cid:44) → M suchthat ω | L = 0. Calabi-Yau manifolds are by definition compact and thus all holomorphicvolume forms on them are locally constant. We can certainly always scale Ω (uniquelyup to phase) such that we have ω n /n ! = ( − n ( n − / ( i/ n Ω ∧ ¯Ω. This scaled Ω is thenthe unique Calabi-Yau form which is a nontrivial calibration on M . For any Lagrangiansubmanifold, L (cid:44) → M , V ol L = e iπθ Ω L for some θ : L → R . We will call this θ , the phaseof L (see discussion in section B.4 below). Definition B.5.
The special Lagrangian submanifolds of M are the Lagrangian submani-folds of M such that θ is constant. Equivalently we could define them to be the submanifolds of M calibrated by Re(Ω)under some choice of phase (i.e. the phase of L ). L | ∝ (cid:81) k ( dx k + idy k ) (cid:54) = 0 if and only if L is aLagrangian submanifold.46Sometimes special Lagrangian submanifolds are defined using a fixed phase. N INTRO TO HMS AND THE CASE OF ELLIPTIC CURVES 41
Example:
By the above discussions of calibrated and minimal submanifolds it is immediatethat all special Lagrangian submanifolds of a (complex) 1-dimensional Calabi-Yau manifold(i.e. an elliptic curve) are geodesics. It is not hard to see that the converse is also true asthe geodesics are exactly the submanifolds of constant phase. Some less trivial examplesand a proof of the following theorem can be found in [44].
Theorem B.6.
A submanifold L of C n = R n is both Lagrangian and minimal if and onlyif L is special Lagrangian. B.4.
Graded Lagrangian Submanifolds. θ is obviously unique only up to an integer shift (and well-defined, mod Z , only with re-spect to a fixed Ω). Given a fixed Ω, in the case where θ is constant (i.e. L is special), we callthis choice the grading of L . A graded special Lagrangian submanifold is then a correspond-ing pair ( L, θ ). This is equivalent to the concept of grading used below in the example ofhomological mirror symmetry for elliptic curves. The shift (
L, θ )[ n ] := ( Lθ + n ) mirrors the n th shift functor on complexes of coherent sheaves. Notice that the shift ( L, θ ) (cid:55)→ ( Lθ + 1)is equivalent to reversing the orientation of L induced by vol L (vol L (cid:55)→ − vol L sends e iπθ → e − iθ ).We can generalize this concept of grading to other Lagrangian submanifolds by definingthe average phase of L , φ ( L ), given modulo Z by (cid:82) L Ω (cid:82) L vol L = e iπφ ( L ) This is clearly invariant under Hamiltonian deformations of L (as this ratio is holomorphicand thus must be constant) and, assuming L is homologous to a special Lagrangian sub-manifold, matches the restricted concept of grading given above. References [1] M.F. Atiyah. Vector Bundles over an Elliptic Curve.
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Department of Mathematics, University of California, Davis, CA 95616–8633, U.S.A.
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