Mirror symmetry for the four-torus

Homological mirror symmetry for the four-torus

Mohammed Abouzaid,  Ivan Smith
August 19, 2019
Abstract.

We use the quilt formalism of Mau-Wehrheim-Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard 4-torus. As an application, we study Lagrangian genus two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of with linear Lagrangian 2-tori in .

This work was conducted during the time the first author was supported by a Clay Research Fellowship. The second author is partially supported by grant ERC-2007-StG-205349 from the European Research Council.

1. Introduction

Despite being the focus of a great deal of attention, Kontsevich’s homological mirror symmetry conjecture [Kon95] has been fully proved in only a handful of cases. In the original Calabi-Yau setting, the elliptic curve was treated by Polishchuk and Zaslow [PZ98, Pol04], whilst Seidel proved the conjecture for the quartic K3 surface [Sei03b]. Substantial but partial results are known for a wide class of abelian varieties, from work of Fukaya [Fu02] and Kontsevich-Soibelman [KS01]. In each of the last two studies, explicit embeddings of (subcategories of) the derived category into the Fukaya category were constructed, but it was not obvious that these embeddings actually induced equivalences. An equivalence of superconformal field theories was separately established for flat tori by Kapustin and Orlov [KO03].

The step required to complete these arguments to a proof of mirror symmetry is therefore to show that the image subcategories generate the entire Fukaya category. Nadler’s paper [Na06], which concerns a certain Fukaya category of Lagrangian submanifolds of the cotangent bundle, has highlighted the importance of the notion of resolutions of the diagonal in proofs of such a result. The idea is that the diagonal (as a Lagrangian in the square of the given symplectic manifold ) represents the identity functor of the entire Fukaya category, so it suffices to show that the diagonal can be decomposed, in an appropriate sense, using products of Lagrangians that one understands. Beyond the technical problem of formalising the connection between Floer theory on a symplectic manifold and on its square, this approach reduces the study of the Fukaya category of a symplectic manifold – which might be teeming with countless unseen Lagrangians – to the study of an explicit, often finite collection of Lagrangians in . In algebraic geometry the corresponding circle of ideas is well-known, going back to Beilinson [Bei78], and underlies Seidel’s split-generation result for Fukaya categories of Lefschetz fibrations by vanishing cycles [Sei08, Remark 18.28].

The main technical observation of this paper is that the theory of pseudo-holomorphic quilts, currently under development by Mau, Wehrheim and Woodward, not only establishes a functor relating the Fukaya category of a product to those of its factors, but moreover, using additional input from homological algebra, reduces the necessary computations on the product manifold to ones which can be performed on the original space. The resulting generative criterion Theorem 7.2, applied to the standard four-torus, completes the proof of homological mirror symmetry in this case. Amusingly, the argument relies only on the proof of Homological Mirror Symmetry for the elliptic curve; we need not appeal to the deeper work of Fukaya [Fu02] or of Kontsevich-Soibelman [KS01], nor perform any (serious) Floer theoretic computation in .

To give a precise statement of the result we prove, let denote the Novikov field

(1.1)

This is an algebraically closed field of characteristic zero. Let denote the Tate elliptic curve, namely the projective algebraic variety over with ring of functions

for series defined by:

For a projective variety , we write for its bounded derived category of coherent sheaves. On the symplectic side, let denote the four-torus with its standard symplectic structure , and write for its split-closed derived Fukaya category (background on -categories is given in the following section).

Theorem 1.1.

There is an equivalence of triangulated categories (defined over the Novikov field) .

Remark 1.2.

As noted above, the proof of the split-generation criterion Theorem 7.2 uses the Mau-Wehrheim-Woodward formalism which constructs -functors between Fukaya categories from counts of quilted Riemann surfaces. That theory is still under construction [MWW, W08], but – by design – is easily formalised and axiomatised; indeed that is one of the theory’s most important and satisfactory features. We give a condensed overview of quilt theory in situations – like that relevant to Theorem 1.1 – in which bubbling is not an issue; the properties that we require are subsumed in the Axiom of Quilted Floer Theory of Section 5. These expected formal properties are assumed throughout this paper.

Remark 1.3.

During the course of the proof, we will see that is split-generated by a meridional and a longitudinal circle on the two-torus, whilst is split-generated by the four two-tori obtained by taking pairwise products of these; this is essentially a “Künneth theorem” for the Fukaya category.

The strategy of proof of Theorem 1.1 is to relate the Fukaya category of the product to the category of functors . Modulo (serious) technical restrictions, the argument should imply homological mirror symmetry for whenever it is known for and themselves. The technical restrictions are largely foundational in nature. By definition, the objects of the Fukaya category are Lagrangian submanifolds of Maslov class zero (decorated with some additional “brane” structure, cf. Section 3). In dimension four, any such submanifold is unobstructed: it bounds no holomorphic disc for generic almost complex structures, just for index reasons111This idea was used by Seidel in setting up and proving mirror symmetry for the quartic surface in [Sei03b].. In higher dimensions, the construction of the Fukaya category relies, in general, on the delicate obstruction theory of Fukaya-Oh-Ohta-Ono [FO3], see also Joyce’s [Joy07]. Whilst that theory is now coming into final shape, the issues that obstruction chains might raise in the quilted setting have not yet been addressed. That is the reason Theorem 1.1 is restricted to four-dimensions. For an aspherical symplectic manifold , denote by the “strictly unobstructed” Fukaya category whose objects are (decorated) pairs comprising a Maslov zero Lagrangian submanifold and an almost complex structure for which bounds no -holomorphic discs. The technology of the current paper carries over rather straightforwardly to prove:

Corollary 1.4.

For any integer , there is an equivalence of categories (defined over the Novikov field) .

When is a power of , there is nothing beyond what we prove in this paper together with induction. Otherwise, the relevant homological algebra is slightly more involved, requiring the use of categories of functors rather than simply of endofunctors. More generally, if one were willing to flesh out the arguments presented by Kontsevich and Soibelman in [KS01], one could conclude a similar result for a wide range of abelian varieties. However, the category is of at best marginal interest from the point of view of symplectic topology, so we will only discuss four-manifolds in the body of this paper, deferring some further speculative remarks to the Appendix.

In fact, proving mirror symmetry was in some sense a subsidiary of our main intention, which was to explore its consequences for symplectic topology. In this vein, our principal result is:

Theorem 1.5.

Let be a Lagrangian genus two surface of Maslov class zero. Then is Floer cohomologically indistinguishable from the Lagrange surgery of some pair of linear Lagrangian tori meeting transversely once.

Although this is a rather specialized result, we should point out that it seems currently inaccessible without mirror symmetry: Lagrangian isotopy or uniqueness theorems in four dimensions typically rely on constructions of holomorphic foliations which do not exist, generically, on the torus, and our proof relies crucially on results of Mukai and Orlov on sheaves on abelian varieties. Despite its rather abstract formulation, Theorem 1.5 has direct implications for intersection properties of Lagrangian submanifolds of ; it imposes numerical restrictions on such intersections reminiscent of those implied by the Arnol’d conjecture.

Corollary 1.6.

Let be a Lagrangian genus 2 surface of Maslov class zero. There are at least two Lagrangian tori (in rationally independent homology classes) with the property that any surface Hamiltonian isotopic to and meeting transversely does so in at least 3 points.

Remark 1.7.

For arising as a Lagrange surgery of a pair of transverse linear Lagrangian tori, one can isotope through Lagrangian non-Hamiltonian isotopies to meet any linear Lagrangian torus at most once.

In the setting of Corollary 1.4, a Maslov zero Lagrangian submanifold must have non-vanishing Floer cohomology with at least one comprising a torus fibre of a Lagrangian fibration equipped with a local system , or must bound a -holomorphic disc for every compatible almost complex structure . This is just the mirror of the fact that no complex of sheaves on an algebraic variety can have vanishing ’s with the structure sheaves of all closed points; but it seems far from obvious by direct symplectic arguments. In another direction, Kapustin and Orlov [KO04] argued that the Fukaya category of a high-dimensional torus did not seem “large enough” (from the point of view of -theory) to be mirror to the derived category of sheaves of an abelian variety, and suggested that one should augment the Fukaya category with certain co-isotropic branes; but Corollary 1.4 indicates that, after passing to split-closures, these are in fact not necessary.

Acknowledgments

Thanks to Daniel Huybrechts, Dima Orlov, Paul Seidel and Nick Shepherd-Barron for helpful suggestions and correspondence, as well as to Sikimeti Mau, Katrin Wehrheim and Chris Woodward for sharing with us a preliminary draft of their paper [MWW]. Detailed comments on our own preliminary draft from many of the same people have saved us from numerous (additional) errors: we are especially indebted to Dima Orlov. Extensive and constructive comments from the anonymous referees have also greatly improved the exposition.

2. -algebra

We begin by collecting some basic facts about -categories. These results were proved by various authors, starting with Kadeishvili [Ka82]. Any discussion of -categories involves fixing sign conventions, of which there are several; for consistency, we have chosen to cite all results from Seidel’s book [Sei08], whose conventions we borrow. The reader entirely unfamiliar with this material will probably find our treatment too cursory, and is invited to consult Part I of Seidel’s book for a more leisurely development, for proofs, and for the appropriate references to the original papers where the results were proved.

2.1. -categories and functors

Fix an arbitrary field . A non-unital -category over comprises: a set of objects Ob; for each Ob a graded -vector space ; and -linear composition maps, for ,

of degree (the notation refers to downward shift by ). The maps satisfy a hierarchy of quadratic equations

with and where the sum runs over all possible compositions: , . The equations imply in particular that is a cochain complex with differential ; the cohomological category has the same objects as but morphism groups are the cohomologies of these cochain complexes. This has an associative composition

but the chain-level composition on itself is only associative up to homotopy. (Thus is not strictly a category.) Moreover, the higher-order compositions are not chain maps, and do not descend to cohomology. With appropriate sign conventions, a -category is just the special case in which for all .

A non-unital -functor between non-unital -categories and comprises a map : Ob Ob, and multilinear maps for

now satisfying the polynomial equations

Any such defines a functor which takes ; if is an isomorphism, respectively full and faithful, we say is a quasi-isomorphism, respectively cohomologically full and faithful. The collection of -functors from to themselves form the objects of a non-unital -category -, whose morphism groups are groups of -natural transformations. Concretely, given two functors and , one defines a pre-natural transformation to be a sequence , with a collection of maps

for all sequences of objects in . The vector space generated by all pre-natural transformations is by definition the space of morphisms between and in . The formula for the differential on can be found, for example, as Equation (1.8) in [Sei08].

The vector space admits a decreasing length filtration with consisting of all pre-natural transformations for which . The associated spectral sequence has as its first page

(2.1)

We should emphasise that there are in general more -functors between -categories than -functors even at the level of homology, so when the higher order operations vanish, there is content to regarding as an -category. In the other direction, Kadeishvili [Ka82] showed:

Lemma 2.1 (Homological Perturbation Lemma).

Any -category is quasi-isomorphic to an -structure on its cohomology

with .

If is quasi-isomorphic to the trivial -structure on its cohomology, namely to the structure with for , we say that is formal.

2.2. -modules

An -category has a category - of right -modules, which abstractly is the category of functors - from (the opposite of) to the -category of chain complexes of graded -vector spaces. Concretely, an -module associates to any Ob a graded vector space , and there are maps for

The -functor equations imply in particular that is the differential on the chain complex . In the sequel, we shall make particular use of the -version of the Yoneda Lemma [Sei08, Lemma 2.12]. To state this, note that for any Ob, there is an associated module defined by

The association extends to a canonical non-unital -functor -, the Yoneda embedding. There is a dual Yoneda embedding into the same category of modules

with the module structure given by

The proof of the next result uses the fact that the spectral sequence (2.1) collapses at the second page to the column because of the acyclicity of the bar resolution:

Lemma 2.2 (Yoneda Lemma).

The natural map taking

is a quasi-isomorphism. The association defines a cohomologically full and faithful functor .

The category of -modules is naturally a triangulated -category: morphisms have cones. More precisely, if is a degree zero cocycle, , there is an -module defined by

(2.2)

and with operations given by the pair of terms

One can generalise the mapping cone construction and consider twisted complexes in : a twisted complex is a pair where is a formal direct sum

(2.3)

with Ob and finite-dimensional graded -vector spaces – i.e. is an object of the “additive enlargement” – and where is a matrix of differentials

(2.4)

with , and having total degree . The differential should satisfy the two properties

  • is strictly lower-triangular with respect to some filtration of ;

  • .

Twisted complexes themselves form the objects of a non-unital -category , which has the property that all morphisms can be completed with cones to sit in exact triangles. A basic example of a twisted complex is that obtained simply by taking the tensor product of an object by the vector space placed in some non-zero degree; in particular, the category has a shift functor, which practically has the effect of shifting all degrees of all morphism groups downwards by one.

A particularly important class of mapping cones are those arising from twist functors. Given Ob and an -module , we define the twist as the module

(with operations we shall not write out here). The twist is the cone over the canonical evaluation morphism

where denotes the Yoneda image of . For two objects Ob, the essential feature of the twist is that it gives rise to a canonical exact triangle in

(where is any object whose Yoneda image is ). Twist functors have played a critical role in relating algebraic properties of Fukaya categories and geometric properties of the underlying symplectic manifold, as briefly indicated below (Proposition 6.1).

2.3. Unitality and projection functors

The -categories we shall consider are cohomologically unital, meaning the categories are unital (objects have identity morphisms, so is a graded linear category in the usual sense). In fact, if one makes careful choices in constructing the Fukaya category, its objects are equipped with lifts of the cohomological units satisfying

From a more formal point of view, we note that the existence of an element satisfying these conditions for an object is equivalent to the existence of an -functor

(2.5)

with vanishing higher order terms. Here, we think of as a category with one object whose endomorphism algebra is the ground field ; the functor takes this unique object to , and its linear term maps to the unit of .

Starting with any finite dimensional co-chain complex and such a linear functor with target , we may naturally define a twisted complex

(2.6)

In the notation of Equations (2.3) and (2.4), the vector spaces are the graded components of , and the differential is only non-vanishing if , in which case it is given by the tensor product of the differential on with the identity on :

The existence of such units also allows us to define certain projection functors which in the general cohomologically unital case only make sense after passing to modules.

Definition 2.3.

For each pair of objects of and with the image of a linear functor with source , we define a projection functor

(2.7)

which acts on objects by

and on morphisms by

(2.8)
(2.9)

One may understand this construction from the abstract point of view by noting that the existence of the twisted complex is part of a higher tensor structure on the category of -categories. As we shall not require the full power of such -categorical machinery, and as the axiomatics of such a structure are rather delicate, we focus instead on a special situation which exploits the fact that the tensor product of an -category and a -category may be easily defined by a formula analogous to (2.9).

In particular, given -categories and , -categories and , and -functors , and such that has no higher order terms, one may define the tensor product of and

whose higher order terms are obtained by applying the higher order terms of .

If is the category with one object, we have a canonical isomorphism , while if is the category of chain complexes over , we have a fully faithful embedding

(2.10)

given by the construction of Equation (2.6). In this language, the functor is isomorphic to the composition of (2.10) with the tensor product of the Yoneda functor for and the functor from Equation (2.5):

Given a pair of objects and such that and are the images of under linear functors, let us write and . As a result of the previous discussion we expect that is quasi-isomorphic to the tensor product of morphisms from to with morphisms from to . To prove this, we consider the functors at the level of homological categories

which are honest functors with no higher order terms. There is a natural map

(2.11)
(2.12)

where is the category of cohomological functors in which morphisms consist of natural transformations. The standard categorical Yoneda argument (rather than an version thereof) allows us to readily compute that

(2.13)
Lemma 2.4.

Every natural transformation between and is detected at the level of homology, i.e. the map (2.11) is an isomorphism. In particular,

(2.14)
Sketch of proof.

The proof is a minor generalisation of that of the Yoneda Lemma. Namely, we consider the length filtration on , and observe that Equation (2.1) specialises, in this case to

(2.15)

Using adjunction, this can be more conveniently rewritten as

(2.16)

Note that this is the tensor product of the page of the spectral sequence computing with the graded vector space . As the differential involves only the product on homology, it is easy to check that it is given by the tensor product of the differential on the page for with the identity on . As the spectral sequence for collapses at the second page to the column , we conclude the same result for . Note that the column precisely consists of pre-natural transformations with non-zero term, i.e. ones which survive the projection to . ∎

2.4. Idempotents and homological invariants

The split-closed (also called idempotent-closed, or Karoubi-complete) derived category of is obtained from by splitting idempotent endomorphisms. Instead of giving the details, we just point out [Sei08] that an -category is idempotent-closed if and only if its cohomological category has the same property, so one can view the passage from to as formally including objects which represent summands of endomorphism rings associated to cohomological idempotents. If the smallest split-closed triangulated -category containing a subcategory is , then we will say that split-generates .

To conclude the background in algebra, we mention two homological invariants of an -category. The first is the K-theory, or rather the Grothendieck group

(2.17)

where we impose a relation whenever is quasi-isomorphic to the mapping cone of a closed degree one morphism . From the definition of twisted complexes, the -group is actually generated by objects of (by contrast its behaviour under passing to split-closure is rather wild in general). Lastly, we also recall the definition of the Hochschild cohomology of an -category . The most concise definition is to view as the morphisms in the -category of endofunctors of from the identity functor to itself. More prosaically, is computed by a chain complex as follows. A degree cochain is a sequence of collections of linear maps

for each . The differential is defined by the usual sum over possible concatenations

(2.18)

In particular, defines an endomorphism of every object of the category. The above formula for the differential readily implies the next result:

Lemma 2.5.

For any object of , the assignment

is a chain map.

Classically, Hochschild cohomology arises in deformation theory. Any (formal, i.e. ignoring convergence issues) deformation of the -structure defines a class in , so for instance if this is one-dimensional the category has a unique such deformation up to quasi-isomorphism. We should also point out the following basic algebraic fact.

Proposition 2.6.

Hochschild cohomology is invariant under taking twisted complexes and under passing to idempotent completion.

There seems to be no written account of this result in the setting of -categories over a field, but the more general result for spectra is Theorem 4.12 of [BM08].

3. The Fukaya category

Let be a closed symplectic manifold and suppose . In ideal situations, the Fukaya category is a triangulated -graded -category, linear over the Novikov field . It has an associated (honest) triangulated category , the split-closed derived Fukaya category; one can also pass directly to cohomology, forgetting the -structure, to obtain the (quantum or Donaldson) category . The objects of the Fukaya category are Lagrangian submanifolds which are decorated with additional data, the existence of which form a collection of strong constraints:

  • the submanifolds should have vanishing Maslov class, and be equipped with gradings [Sei00];

  • the submanifolds should be spin, or relatively spin relative to a fixed background class (though not strictly necessary we moreover only consider orientable Lagrangians);

  • the Floer cohomology of the submanifolds should be unobstructed for some choice of bounding chains in the sense of [FO3].

Moreover, the Fukaya category should have the properties that

  • Hamiltonian isotopic Lagrangian submanifolds define isomorphic objects of ;

  • up to quasi-equivalence is a symplectic invariant of .

The final statement is intentionally vague: one expects a canonical map , where is a natural subgroup of symplectomorphisms which preserve the structure needed to grade the category, e.g. the homotopy class of trivialisation of , and on the right hand side we divide out by the shift functor. As indicated in Section 2, the construction of the derived category from is a purely algebraic procedure.

Remark 3.1.

The Novikov field comprises formal sums

(3.1)

The Fukaya category has higher-order -operations defined by counts of certain pseudoholomorphic polygons: these counts assemble into power series which are not known to have positive radius of convergence, so the category is only well-defined over a field of formal power series. It is also possible to allow as objects of Lagrangian submanifolds equipped with flat (typically unitary) line bundles; we will not need this extension, but see Remark 6.7.

The first of the conditions imposed on objects of – vanishing of the Maslov class – makes sense whenever , and enables the category to be -graded. The second condition – existence of (relative) spin structures – enables one to coherently orient the moduli spaces of pseudoholomorphic polygons entering into the definitions of the and hence define the category with coefficients in a field not of characteristic 2. (In the absence of spin structures, one should take in Equation 3.1.) The third condition is required for the endomorphisms of an object to actually be well-defined. Recall that Floer cohomology for a pair of Lagrangian submanifolds is defined (roughly) as follows. One picks a Hamiltonian flow for which is transverse, takes , and defines a differential which counts solutions to a perturbed Cauchy-Riemann equation: , where is the space of solutions to the perturbed Cauchy-Riemann equation

(3.2)

with boundary and asymptotic conditions

These holomorphic strips come in moduli spaces which, for suitably generic families of compatible almost complex structures, are manifolds, and one counts the zero-dimensional components. Morphisms in the Fukaya category are by definition the Floer chain groups; the higher order operations of the -structure comprise a collection of maps of degree , for , with being the differential:

These have matrix coefficients which are defined by counting holomorphic discs with -boundary punctures, whose arcs map to the Lagrangian submanifolds in cyclic order. To be slightly more precise, to each intersection point one actually associates the group of coherent orientations – freely generated by the two orientations of a one-dimensional vector space, subject to the relation that their sum vanishes. When the Lagrangian submanifolds are relatively spin, the moduli spaces of pseudoholomorphic polygons carry determinant lines which define orientations relative to these coherent orientation spaces, i.e. which yield isomorphisms

see for instance [Sei08, Section 12b]. It follows that the isolated points of moduli spaces carry canonical signs relative to the orientation groups associated to intersection points, and the counts are really signed counts if we work in characteristic zero.

To achieve transversality for the moduli spaces of holomorphic strips, one replaces a fixed compatible almost complex structure in Equation (3.2) by a family depending on the second factor of a strip. More generally, we replace by families of almost complex structures indexed by points of the abstract underlying holomorphic disc; such “universal consistent choices of perturbation data” [Sei08] are constructed inductively over the moduli spaces of holomorphic discs (associahedra). The resulting maps are not chain maps, and hence do not naively descend to Floer homology; they are chain-level operations which satisfy the hierarchy of quadratic associativity equations which define an -structure:

Crucially, the individual count comprising a given matrix element amongst a particular collection of Lagrangian submanifolds is not well-defined (independent of perturbations or Hamiltonian isotopy), but the entire -structure is well-defined up to quasi-isomorphism.

Proposition 3.2.

If bounds no holomorphic discs for some compatible almost complex structure, then .

This goes back to Floer [Flo88]. Fixing a Morse function on defines a Hamiltonian perturbation by the associated Hamiltonian flow of the vector field . The generators of the Floer complex correspond bijectively to critical points of , and in the absence of bubbling and by a judicious choice of time-dependent almost complex structure, Floer identified the complex with the Morse complex of . However, if there are holomorphic discs with boundary on or , one can lose control of the compactness of the moduli spaces of perturbed holomorphic strips in a way which breaks the -equations, and even breaks the first such . Floer homology is said to be “unobstructed” if one can make choices (in general infinitely many, of a delicate inductive nature) to repair this basic deficiency. The choices of “bounding chains” on “cancel” the errant disc bubbles. Rather than grapple with the deep material of [FO3], where their properties are addressed in great generality, we opt for a rather low-brow alternative: Floer cohomology is (trivially) unobstructed if for some compatible almost complex structure there are no holomorphic spheres passing through , and the moduli spaces of holomorphic discs with boundary on are actually empty.

Lemma 3.3.

Let be a four-dimensional symplectic manifold with . For a generic almost complex structure compatible with and a Lagrangian submanifold of Maslov class zero, there are no -holomorphic discs with boundary on or spheres passing through .

Proof.

A -holomorphic sphere or disc is somewhere injective if there is a point for which and . If is compact without boundary, any simple map (one which does not factor through a branched cover of over another curve) contains a dense set of somewhere injective points [McD87]. For curves with boundary, a theorem of Kwon-Oh [KO00] and Lazzarini [Laz00] implies the weaker statement that if bounds some -holomorphic disc, then it bounds a simple disc. In both cases, transversality of the Cauchy-Riemann equation can be achieved at simple curves by choosing a generic almost complex structure on .

If is a -dimensional symplectic manifold with , the Riemann-Roch theorem for curves with boundary gives the dimension of the space of unparametrised discs with boundary on to be . If the Maslov class and this is negative, hence for generic the moduli spaces of simple discs are actually empty. The result of Kwon-Oh or Lazzarini then implies that bounds no holomorphic discs at all, for generic . The same argument shows that when the symplectic -manifold contains no holomorphic spheres for generic . ∎

Although the Floer Equation (3.2) really involves families of almost complex structures , the bubbles that obstruct are honest holomorphic discs (for respectively if the bubble appears on the lower respectively upper edge of the strip). Lemma 3.3 accordingly implies that the Fukaya category of a four-dimensional symplectic Calabi-Yau manifold can be defined, and any Lagrangian surface with vanishing Maslov class defines a non-zero object of this category (Proposition 3.2 shows that it has non-trivial endomorphisms). However, it is not a priori obvious that Hamiltonian isotopic Lagrangian surfaces define isomorphic objects, because in a one-parameter family of Lagrangian submanifolds one does in general expect to encounter bubbles. In some situations, bubbles can be excluded for topological reasons.

Lemma 3.4.

Let be a four-dimensional symplectic manifold and a Lagrangian surface. If the map vanishes then the isomorphism class of in depends only on the Hamiltonian isotopy class of .

Proof.

It suffices to show that no Hamiltonian image of bounds any (non-constant) holomorphic disc. The symplectic form defines an element and the area of a holomorphic disc is given by the pairing between the symplectic form and the image of in . The result follows. ∎

For instance, for Lagrangian submanifolds whose relative vanishes, Floer cohomology is defined unproblematically in any dimension and the Fukaya isomorphism type is unchanged by Hamiltonian isotopy. It follows that linear tori are always elements of the “strictly unobstructed” Fukaya category whose objects are, by definition, Lagrangian submanifolds which for some compatible bound no holomorphic discs and are intersected by no -holomorphic spheres. These are the only objects we require when dealing with Fukaya categories of higher-dimensional tori in the rest of the paper.

Fukaya categories are cohomologically unital but not strictly unital; on the other hand, the cohomological units have geometrically meaningful chain-level representatives.

Lemma 3.5.

The -category can be equipped with distinguished elements which are cycles whose cohomology classes are the units in .

Proof.

Given , we have picked a (time-dependent) Hamiltonian function for which . We can regard the generators of the Floer complex as the time-1 chords from to itself under the Hamiltonian flow of . The unit element is obtained as the count of rigid solutions to the perturbed Floer equation with domain a disc with one boundary puncture and boundary condition the family . It is well-known that this represents the cohomological unit, by a standard application of gluing. ∎

Remark 3.6.

Morally, counts rigid finite-energy half-planes with boundary on ; if, following Joyce [Joy07], we defined to be the space of Kuranishi chains on , then this would be precisely true.

Lemma 3.7.

The perturbation data for the -structure can be chosen such that

Proof.

As in the discussion after Proposition 3.2, we can define by choosing a Hamiltonian perturbation of arising from a Morse function on . The Floer complex is then concentrated in degrees , from which the second equation follows immediately since the -operation has degree . Taking the perturbing Morse function to have a unique maximum and minimum on each connected component of implies that the first equation also holds. ∎

Remark 3.8.

Although there is some formal diffeomorphism making strictly unital, this will in general not be geometric (i.e. the particular matrix entries defining the strictly unital structures will not actually be counts of holomorphic polygons for any choice of perturbation data). The condition of Lemma 3.7 is much weaker than strict unitality since we are only constraining the behaviour of higher products where all the inputs are the identity.

Once one has set up the Fukaya -category, one can appeal to the general machinery of Section 2 to construct categories of twisted complexes, etc. There are certain situations in which the algebraic operations defining twisted complexes correspond precisely to geometric operations amongst Lagrangian submanifolds.

Example 3.9.

Suppose is a Lagrangian sphere. Then given any Lagrangian submanifold , one can form either the geometric Dehn twist , or the algebraic twist which is (quasi-represents) the cone over the canonical evaluation . These are actually quasi-isomorphic objects of , by a theorem of Seidel [Sei08].

Suppose and are oriented Lagrangian submanifolds of meeting transversely in a single point . The Lagrange surgery is a Lagrangian submanifold smoothly isotopic to the connect sum of the (topologically there are two such local surgeries, only one of which is compatible with the fixed local orientations). The four-dimensional case goes as follows [Sei99]. Order the and choose a Darboux chart near the intersection point which linearises the ordered pair to the (oriented) Lagrangian planes . If is a smooth embedded curve which lies in the lower right hand quadrant and co-incides with the positive -axis union the negative -axis outside a sufficiently small ball near the origin, the Lagrange handle is defined by

This is diffeomorphic to , co-incides with outside a compact set, and is Lagrangian for the standard symplectic form on . The Lagrange surgery is obtained by replacing