Mirror symmetry and deformation quantization

arXiv:hep-th/0202128v3 19 Apr 2002 Mirror symmetry and deformation quantization Paul Bressler, Yan Soibelman February 1, 2008 1 Introduction In homological mirror symmetry (see [Ko1]) and in the theory of D-modules one meets similar objects. They are pairs (L, ρ) where L is a Lagrangian manifold, and ρ is a flat bundle on L (local system). In the framework of homological mirror symmetry such pairs are objects of the so-called Fukaya category, which is the principal mathematical structure of the genus zero part of the A-model. In the framework of D-modules they are holonomic Dmodules (in C ∞ category). It is natural to compare the categories themselves. This comparison is the main theme of present paper. One can object any relationship between mirror symmetry and deformation quantization. Let us mention some of possible objections. a) Theory of D-modules (or more generally modules over the ring of microdifferential operators) “lives” on the cotangent bundle T ∗ X, while the Fukaya category is defined for any symplectic manifold, hence is more general. b) Theory of D-modules works well in algebraic or complex analytic framework, while the Fukaya category (and in general Floer theory) exists in C ∞ category only. c) In mirror symmetry one considers series in exponentially small (with respect to the symplectic structure) parameter, while in deformation quantization the parameter “is of the size” of the symplectic structure. From our point of view these are rather problems than objections. In regard to the point a) this means that one should work out a version of 1 the theory of microdifferential operators for arbitary symplectic manifolds. This theory is known as deformation quantization of symplectic manifolds. In regard to the point b) we remark that finding of a complex analog of the Floer theory is an interesting problem. At this time we have only speculations in this directions. The aim of present paper is to summarize our present understanding of the topic and make some conjectures. It is a part of an ongoing project. The details will appear elsewhere. Our main conjecture can be briefly formulated such as follows: i) there is a category of holonomic modules over the quantized algebra of smooth functions on a symplectic manifold; ii) it is possible to change morphisms in this category by a kind of integral transformation, so that it becomes equivalent (at least locally) to the Fukaya category of the same symplectic manifold. Let us mention two possible applications of this idea. First, it can help in constructing of an algebraic model of the Fukaya category (analogy: de Rham complex is an algebraic model of Morse complex). Second, it might help to resolve some difficulties in the definition of the Fukaya category (nontransversality of supports, non-existence of identity morphisms, etc.) In our opinion current situation is not completely satisfactory despite the recent progress (cf. for ex. [FOOO]). Third, one can go beyond Lagrangian submanifolds by considering coisotropic submanifolds or even non-smooth varieties(cf. [KO]). Finally, one hopes to achieve a deeper understanding of the relationship at the level of chiral algebras (see [BD],[MSV], [KaV]). Hence the subject of this paper is just the first approximation to the full picture. We hope to discuss it elsewhere. The paper is organized as follows. In Section 2 we briefly recall basics on the deformation theory and A∞ -categories. The purpose of this section is to fix the language. Details of the formalism will be explained in [KoSo2]. In Section 3 we recall main facts about the Fukaya category, which is one of the main structures of homological mirror symmetry. Section 4 is a reminder on deformation quantization of symplectic manifolds. Section 5 is devoted to the comparison of the Fukaya category with the category of holonomic modules over the quantized algebra of functions. Main idea of Section 3 goes back to the pioneering paper [Ko1]. As explained in loc. cit. homological mirror symmetry is not a statement about individual categories, but rather about families of A∞ -categories over Z-graded formal schemes. Geometrically this structure is modelled by a “family of non2 commutative differential-graded manifolds over a commutative differentialgraded base”. The framework in which the latter phrase has precise meaning will be explained in [KoSo2]. The reader can keep in mind the example of the deformation theory of an associative algebra. In this case one has a family of associative algebras parametrized by the formal moduli space M of deformations of the given algebra A. The moduli space M is a formal pointed differential-graded manifold (dg-manifold for short) (see for ex. [Ko2], [KoSo3]). Then by definition M is a base of a family of associative algebras Aγ , γ ∈ M. Each algebra gives rise to a “non-commutative scheme” Spec(Aγ ), hence one has the desired structure. Acknowledgements. We thank to Dima Arinkin and especially to Maxim Kontsevich for useful discussions. Some of the background material is taken from the joint book of Kontsevich and second author (see [KoSo2]). We thank to Sergei Barannikov for comments on the paper. Y.S. thanks to the Clay Mathematics Institute for supporting him as a Fellow and to IHES for hospitality and excellent research conditions. He also thanks to the organizers of the workshop CATS-1 (Nice, November 2001) where the main ideas were reported. 2 Reminder on A∞ -categories and deformation theory In this section we recall some facts about homological algebra of mirror symmetry. Full details including the necessary language of non-commutative geometry will appear in [KoSo2]. Some of the material can be found in existing literature, for example in [Ko2]. We are not going to discuss in detail motivations for all of the definitions and notions below. Main purpose of this section is to fix the language. Let A be a free Z-graded k-module over a unital commutative ring k of characteristic zero (main applications deal with the case when k is a field). Definition 1 An A∞ -algebra A over k is given by the following data: a) A Z-graded free k-module A. b) A codifferential d on the cofree coalgebra T (A[1]) = ⊕n≥1 A[1]⊗n , where A[1] denotes the graded free k-module such that A[1]i = Ai+1 (A with shifted grading). 3 (We recall that a codifferential means a coderivation d of the coalgebra satisfying the condition d2 = 0). Since d is uniquely defined on cogenerators, it gives rise to “higher multiplications” mn : A⊗n → A, n ≥ 1 of degrees 2 − n satisfying a system of quadratic equations which follows from the equation d2 = 0. Definition 2 An L∞ -algebra on A is given by the following data: a) A Z-graded vector space A. b) A codifferential on the cofree cocommutative coalgebra C(A[1]) = ⊕n≥1 S n (A[1]), where S n (V ) denotes the nth symmetric power in the symmetric monoidal category of Z-graded k-modules. The codifferential d defines a sequence of “higher Lie brackets” mn : A⊗n → A, n ≥ 1 of degrees 2 − n satisfying a system of quadratic equations which follows from the equality d2 = 0. It is useful to have in mind geometric picture for both algebraic structures defined above. We start with L∞ -algebras. An L∞ -algebra gives rise to a formal pointed Z-graded manifold X, which carries a vector field dX of degree +1 such that dX vanishes at the marked point, and satisfies the condition [dX , dX ] = 0. This structure is called formal pointed differential-graded manifold in [Ko2], [KoSo2], [KoSo3] (it was introduced by A. Schwarz under the name of Q-manifold). Algebra of formal functions of X is isomorphic to the graded dual to the coalgebra C(A[1]). Thus we can write X = Spf ((C[1])∗), where Spf stands for the formal spectrum. Remark 1 a) One should remember that formal Z-graded manifolds have only nilpotent points. b) It is useful to interpret maps mn as Taylor coefficient of the vector field dX at the marked point. Let us consider two illustrating examples. Example 1 Let A be an associative algebra. Then its truncated Hochschild complex C+• (A, A)[1] = ⊕n≥1 Homk (A⊗n , A)[1] carries a structure of differentialgraded Lie algebra, hence defines a formal pointed dg-manifold (see [Ko2]). 4 One can interpret the DGLA from the example as a DGLA of coderivations of the tensor coalgebra cogenerated by A[1]. Equivalently, it is the DGLA of vector fields on the formal pointed graded manifold X vanishing at the marked point. Then the DGLA structure is the natural one on vector fields. One also has the notion of formal differential-graded manifold, where the condition of vanishing at the marked point is dropped. Algebraically this means that we allow the condition m0 6= 0. Example 2 In the previous example we consider the full Hochschild complex C • (A, A)[1] = ⊕n≥0 Homk (A⊗n , A)[1]. It gives rise to a formal dg-manifold. Importance of L∞ -algebras and formal dg-manifolds in deformation theory is based on the fact that they define deformation functors (see for ex. [Ko1], [KoSo2]). Let us briefly recall the construction. If g = ⊕n≥0 g n is an L∞ -algebra (g n is the nth graded component) then one has the deformation functor from commutative nilpotent algebras (possibly graded) to groupoids. Namely to a commutative nilpotent ring R one assigns the groupoid Defg (R) consisting for γ ∈ g 1 ⊗ R which satisfy the Maurer-Cartan equation m1 (γ) + m2 (γ, γ)/2! + ... + mn (γ, ..., γ)/n! + ... = 0, where mn are the higher Lie brackets. Formal deformations of many algebraic and geometric structures give rise to formal pointed dg-manifolds. Formal dg-manifolds without marked points arise, for example, when one deforms categories (i.e. when objects of a category are deformed). In Example 2 the corresponding dg-manifold controls deformations of the category with one object X such that Hom(X, X) = A. One can develop a similar geometric language for A∞ -algebras. Namely, an A∞ -algebra gives rise to a non-commutative formal pointed dg-manifold. It is modelled by the cofree coalgebra T (A[1]) which carries a codifferential dX . One also has the notion of non-commutative formal dg-manifold (no marked point is specified). In this case one can have a non-zero map m0 : k → A. Geometrically this structure corresponds to a vector field dX of degree +1 such that [dX , dX ] = 0, without any condition at the marked point. The corresponding algebraic structure is defined in the following way. Definition 3 We say that a codifferential on the coalgebra k ⊕ T (A[1]) defines a structure of generalized A∞ -algebra on A. 5 The notion of a small A∞ -category is a natural generalization of the notion of A∞ -algebra. Traditionally, such a category is defined by a set of objects Ob(C), Z-graded free k-modules of morphisms Hom(X, Y ), and structures of A∞ -algebras on the spaces ⊕0≤i,j≤n Hom(Xi , Xj ) for any collection of objects X0 , ..., Xn , n ≥ 1. These structures are given by the higher compositions mn : ⊗0≤i≤n Hom(Xi , Xi+1 ) → Hom(X0 , Xn ), which are maps of Z-graded free k-modules of degrees 2−n satisfying quadratic relations similar to those for A∞ -algebras. The structures of A∞ -algebras are compatible with inclusions of collections of objects. One can think of an A∞ -category C as of the large A∞ -algebra End(⊕X∈Ob(C) X) (compare with the relation between additive categories and associative algebras). The Hochschild complex and Hochschild cohomology of an A∞ category can be defined in terms of this A∞ -algebra. For any object X the k-module End(X) = Hom(X, X) is an A∞ -algebra. Its truncated Hochschild complex gives rise to a formal pointed dg-manifold MX . Then the formal dg-manifold M = ⊔X∈Ob(C) MX “controls” A∞ deformations of the category C with the fixed set of objects. Replacing in the above discussion the truncated Hochschild complex by the full Hochschild complex one obtains a new structure called generalized A∞ -category. Generalized A∞ -categories do not have a fixed set of objects. This is due to the fact that now for the A∞ -algebra End(X) one can have m0 6= 0. As before, one can derive the formal dg-manifold M (now using the generalized A∞ -algebras End(X)). For a commutative nilpotent k-algebra R one can consider k-points M(R). If M0 ⊂ M is the subset of zeros of the odd vector field dM then one can speak about objects of some A∞ -category. The objects are parametrized by M0 . We omit here the description which can be given in terms of R-points of M0 and M. We will keep the name of generalized A∞ -category for a slightly more general structure. The point is that we allow the higher compositions mn be defined not for all collections of objects, but only for some of them (transversal collections). The Fukaya category discussed in the next section will be this kind of generalized A∞ -category. We summarize without details the data defining a generalized A∞ -category in the following way: a) We are a given a formal dg-manifold Ob(C) = M. b) For any n ≥ 1 we are given a formal dg-submanifold Mntr ⊂ Mn called 6 the space of transversal n-families. It is assumed that M1tr ⊂ M (i.e. every object is transversal to itself). c) For a point (X, Y ) ∈ M2tr we are given a Z-graded free k-module HomC (X, Y ) called a space of morphisms between X and Y . These kmodules are organized in a formal dg-bundle HomC → M2tr . d) For n ≥ 1 and any (X0 , ..., Xn ) ∈ Mntr we are given a higher composi(X ,...,Xn) tion map mn 0 which gives rise to a morphism of the obvious pullbacks C n of Hom to Mtr . The composition maps give rise to a structure of generalized A∞ -algebra A(X0 , ..., Xn ), or, equivalently to a non-commutative formal dg-manifold M(X0 , ..., Xn ). e) Let M(C) be the inductive limit of M(X0 , ..., Xn ) taken over increasing collections of transversal objects. This is a non-commutative formal dg-indscheme (it can be properly defined as an inductive limit in the appropriate category). f) Finally M(C) is a non-commutative formal dg-manifold over the commutative scheme Spec(k). The structure defined in a)-e) is called generalized A∞ -category. It gives rise to the usual k-linear A∞ -category, which we will call associated to C. The latter is defined by a non-commutative formal dg-ind-subscheme M(C)0 ⊂ M(C) of zeros of the odd vector field dM(C) . Remark 2 a) One can define the Hochschild complex of a generalized A∞ category. It gives rise to a formal dg-manifold. The corresponding deformation functor describes the formal deformation theory of the generalized A∞ -category in the class of generalized A∞ -categories. If the category has one object X we have the full Hochschild complex of the corresponding generalized A∞ -algebra End(X). b) We do not discuss the delicate problem of unital A∞ -algebras (more generally, A∞ -categories with identity morphisms). It is an interesting question because in practice identity morphisms may be defined up to a homotopy only. All this can be formulated in the language of non-commutative geometry. There is a notion of A∞ -functor between two generalized A∞ -categories. A∞ -functors form themselves a generalized A∞ -category. Using A∞ -functors one defines the notion of equivalence of A∞ -categories (see [KoSo2]). Let us illustrate the notion of equivalence in the case when both categories have only one object. Then we are dealing with generalized A∞ -algebras, 7 B say A and B. Assume in addition that mA 0 = 0 and m0 = 0 (i.e. we have ordinary A∞ -algebras). If A is equivalent to B then the complexes B (A, mA 1 ) and (B, m1 ) are quasi-isomorphic. Geometrically this means that the tangent spaces at marked points of equivalent non-commutative formal pointed dg-manifolds are quasi-isomorphic. The converse is also true (this is the A∞ -version of the inverse function theorem). A generalized A∞ -algebra with m0 6= 0 is equivalent to one which has m0 6= 0, and mn≥1 = 0 (cf.: vector field which is non-trivial at a point is locally equivalent to a constant one). The latter observation explains why generalized A∞ -categories should be studied in families rather than individually. Indeed generalized A∞ -categories with m0 6= 0 are trivial in the sense that all higher compositions can be killed by an appropriate equivalence functor. Similarly to the case of formal pointed dg-manifolds there is a theory of minimal models of non-commutative formal pointed dg-manifolds. For such a theory one needs to assume that the ground ring k is a field of characteristic zero (there is more complicated theory for non-pointed dg-manifolds). If two A∞ -categories are equivalent then the formal pointed dg-manifolds of their deformations are quasi-isomorphic (i.e. tangent complexes at the marked points are quasi-isomorphic). Finally, there is a theory of generalized A∞ -categories over a formal dgbase. In the above discussion we discussed the case when the base was an ordinary scheme Spec(k). We can also assume that the base is a formal scheme. As we will see in the next section the latter case is important in symplectic geometry. 3 3.1 Fukaya category Fukaya category and non-commutative geometry The Fukaya category F (X) of a smooth symplectic manifold X is a generalized A∞ -category over a base. It can be constructed as an A∞ -deformation of the following trivial C-linear category F0 (X). Objects of F0 (X) are pairs (L, ρ), where L is a Lagrangian submanifold of X and ρ is a local system on L. Transversal collections of objects correspond to transversal collections of Lagrangian submanifolds (in fact we need more sofisticated transversality condition, see [KoSo1]). We set HomF0 (X) ((L0 , ρ0 ), (L1 , ρ1 )) = ⊕x∈L0 ∩L1 Hom(ρ0x , ρ1x ) and HomF0 (X) ((L, ρ), (L, ρ)) = Ω• (L, End(ρ)). All 8 compositions mn , n ≥ 1 are trivial for collections of different objects. Otherwise mn≥3 = 0, and m2 is the natural product on the differential-graded algebra Ω· (L, End(ρ)). To a pair (L, ρ) one can associate a generalized A∞ -algebra A(L, ρ). Let assume for simplicity that ρ is a trivial rank one local system. Then the corresponding generalized A∞ -algebra A(L) is generated by geometric cycles in L. Higher multiplications mn (C1 , ..., Cn ) between generic cycles Ci are given by a kind of quantum cohomology construction. Namely, one counts with some weight pseudo-holomorphic maps f : (D 2 , ∂D2 ) → (X, L) with marked points x1 , ..., xn ∈ ∂D2 . Here D 2 ⊂ C is the standard Rdisc. It is required that the point xi is mapped to Ci . The weight is exp(− 1ǫ D2 f ∗ (ω)), where ω is the symplectic form on X, and ǫ is a parameter. The idea of this construction of A(L) was suggested by Kontsevich. Difficult analytic details have been worked out in [FOOO]. In what follows we will assume the conditions on the data imposed in [FOOO]. The resulting generalized A∞ P ≥0 −λi /ǫ algebra A(L) is defined over the valuation ring Cǫ = {f = i≥1 ai e }, where ai ∈ C, a1 6= 0, and λi ≥ 0 is a monotonically increasing sequence of real numbers such that limi→+∞ λi = +∞. The valuation map is given by v(f ) = λ1 . The corresponding valuation field Cǫ consists of series f as above, with λi ∈ R. It is useful to notice that C≥0 ǫ contains the maximal ideal C>0 consisting of series with all λ > 0. One observes that the composition i ǫ >0 m0 = 0 modulo Cǫ , but in general m0 6= 0. It can be proved that the disjoint union of non-commutative dg-manifolds associated with generalized A∞ -algebras A(L, ρ) gives rise to a generalized A∞ -category Fǫ (X) over the formal spectrum of the ring C≥0 ǫ . Its reduction modulo the ideal C>0 is equivalent to F (X). If the symplectic form ω sat0 ǫ isfies certain rationality conditions then one can introduce a new parameter q = exp(− 1ǫ ), thus replacing the ring C≥0 ǫ by the ring of formal series C[[q]]. Then the maximal ideal is just qC[[q]], and the field Cǫ coincides with the field of Laurent series C((q)). The Fukaya category F (X) is defined as an A∞ -category obtained from Fǫ (X) by restriction to zeros of the odd vector field. In particular, for an obA ject A of F (X) the composition m0 vanishes: mA 0 = 0. The condition m0 = 0 C defines a “subvariety” of the non-commutative moduli space MN Ob(Fǫ (X)) . Objects of the Fukaya category exist only along this “subvariety”. This geometric picture explains why it is too naive to work with the Fukaya category for fixed ǫ, even if one can prove convergence of the series defining mn (the latter is still an open problem). 9 Let us assume for simplicity the above-mentioned rationality conditions of ω. Then the Fukaya category is defined over the formal spectrum Spf (C[[q]]). In fact one can extend the definition so that the base will be Spf (C[[q]] ⊗ (⊗i6=2 C[ti,µ ])), where q has degree zero, and ti,µ are parameters of degrees 2 − i corresponding to some basis in the graded vector space of cohomology H i(X, C). We introduce new parameters z, t of degree zero by setting z = qet . Then, inverting z we obtain a family of A∞ -categories over the field C((z)) parametrized by Spf (C[[t]] ⊗ (⊗i6=2 C[ti,µ ])). This family should be thought of as formal deformation of a certain A∞ -category over C((z)). The tangent space to the moduli space of the formal deformations of this category is isomorphic to the cohomology H • (X, C((z))). The latter cohomology group is isomorphic to ⊕i≥0 Exti (Id, Id), where Id is the identity functor, and the extensions are taken in the properly defined A∞ -category of endofunctors. This description is useful for the purposes of quantum cohomology (Yoneda product on functors gives rise to the quantum product on the cohomology group). 3.2 Conventional approach to the Fukaya category Below we briefly recall the “naive” definition of the Fukaya category, when the composition m0 is ignored. It is useful in some questions, for example in mirror symmetry for abelian varieties (see [KoSo1]). As we will discuss below, this “naive” Fukaya category is related to deformation quantization. Objects of F naive (X) are pairs (L, ρ), where L ⊂ X is a Lagrangian submanifold and ρ is a local system on L. Morphisms between (L0 , ρ0 ) and (L1 , ρ1 ) are defined only if L0 and L1 intersect transversally. In this case Hom((L0 , ρ0 ), (L1 , ρ1 )) = ⊕x∈L0 ∩L1 Hom(ρ0x , ρ1x ) ⊗ Cǫ . The space of morphisms is Z-graded by means of Maslov index. Thus we are dealing with graded Lagrangian manifolds (cf. [Se1]). There are higher compositions mn , n ≥ 1, which are linear maps of degrees 2 − n: mn : ⊗0≤i≤n Hom((Li , ρi ), (Li+1 , ρi+1 )) → Hom((L0 , ρ0 ), (Ln , ρn )). They are defined by means of the Floer-type construction associated with the “transversal” collection of Lagrangian submanifolds Li , 0 ≤ i ≤ n. It is usually said that the maps mn give rise to an A∞ -structure on F naive (X). We refer the reader to [KoSo1] about the details of this definition and to [FOOO] about definitions of the related moduli spaces. 10 There are several problems with the naive definition of the Fukaya category. One of the most essential is the presence of pseudo-holomorphic discs with the boundary mapped to a Lagrangian submanifold. This amounts to non-trivial maps m0 : Cǫ → Hom(X, X). As a result, the axioms of A∞ -category are not satisfied, and one has to work with generalized A∞ categories. It was explained in the preceeding sections (for all details see [KoSo2]) how a consistent theory of such can be developed in the framework of non-commutative formal geometry. One can still work with F naive (X) with understanding that it is only a part of the “true” Fukaya category F (X). 4 Reminder on deformation quantization We recall that a symplectic 2n-dimensional manifold (X, ω) gives rise to an abelian category C(X) of modules over a non-commutative algebra A(X) (deformation quantization of the algebra C ∞ (X) of smooth functions on X). Such a deformation quantization is non-unique. We will use the one which has the characteristic class [ω]/t (see [De]). The algebra A(X) is a topological algebra over the ring of formal series C[[t]]. As C[[t]]-module it is isomorphic to the algebra of formal series C ∞ (X)[[t]]. The algebra A(X) consists of global sections of a sheaf of non-commutative algebras AX , such that locally AX is isomorphic to the sheaf P of t-pseudo-differential operators n on R (the latter are locally series P = |I|≥0 aI (x)(t∂x )I ). The Poisson ∞ structure induced on CX ≃ AX /tAX coincides with the one given by the symplectic form. One has a category of A(X)-modules M such that M is t-adically complete, flat as C[[t]]-module, and M/tM is the space of sections of a sheaf ∞ of modules over the sheaf of smooth functions CX . The category of AX modules will be denoted by C(X). Morphisms are defined as C[[t]]-linear homomorphisms of topological modules. We will keep the same notation for the related category defined over the field C((t)). It is obtained from C(X) respectively CX by adding t−1 , so that modules V and tV become equivalent. Let hol(X) be a full subcategory of C(X) which consists of modules M such that the support Supp(M/tM) is a Lagrangian submanifold. We will call objects of hol(X) holonomic. The Lagrangian support Supp(M) of a holonomic module will be sometimes called its characteristic variety of M and denoted by Ch(M). The category hol(X) contains objects V(L,ρ) which correspond to pairs (L, ρ) where L ⊂ X is a Lagrangian submanifold and ρ 11 is a local system on L. In what follows only such objects will be considered. Remark 3 In [KaMa] the authors constructed (for every Lagrangian submanifold L satisfying some topological conditions) an AX -module VL such that Ch(VL ) = L. One can easily generalize their construction including local systems on L. We will call the corresponding objects Karasev-Maslov modules. We will need symplectic manifolds X n , n ≥ 2. The corresponding symplectic forms are given by (ω, −ω, −ω, ..., −ω). The identity functor IdAX −mod is represented by the AX×X -module K∆ supported on the diagonal ∆ ⊂ X × X. It can be identified with the sheaf AX . Deformations of AX − mod as an A∞ -category has the tangent complex quasi-isomorphic (after a shift) to the tangent complex to the deformations of the identity functor IdAX −mod . The latter deformations are described by the deformations of K∆ as an object of AX×X − mod. The tangent space at K∆ to the moduli space of its deformations is isomorphic to ⊕i≥0 ExtiAX×X −mod (K∆ , K∆ ). After changing scalars to C((t)) the latter sum can be identified with H • (X, C((t))). We can restrict the deformation functor to the subcategory hol(X). It is not difficult to see that the support of a holonomic module remains Lagrangian. These observations lead to the following result. Proposition 1 a) The Hochschild cohomology of the category AX − mod is isomorphic to H • (X, C((t))). (Here we consider AX −mod as an A∞ -category with mn6=2 = 0 and m2 given by the usual composition of morphisms). b) The tangent space to the deformations of an object (L, ρ) ∈ hol(X) is isomorphic to H • (L, End(ρ)) ⊗ C((t)). We also mention the following proposition (see [So1]). It should be compared with the definition of Hom′ s in the Fukaya category. Proposition 2 Let V(L,ρ) denotes the object of holX correspodning to the pair (L, ρ). If L0 is transversal to L1 then a) Exti (V(L0 ,ρ0 ) , V(L1 ,ρ1 ) ) is trivial if i 6= n, where n = 1/2 dim X; b) Extn (V(L0 ,ρ0 ) , V(L1 ,ρ1 ) ) ≃ ⊕x∈L0 ∩L1 Hom(ρ0x , ρ1x ) ⊗ C((t)); c) the algebra Ext• (V(L,ρ) , V(L,ρ) ) is isomorphic to the cohomology H • (L, End(ρ))⊗ C((t)) (cf. part b) of the previous Proposition). 12 b Let D∞ (hol(X)) be the A∞ -category associated with the category hol(X). It is in fact a dg-category. In order to construct it one choses injective resolutions of AX -modules IM and IN of two holonomic modules M and N. • Then one defines HomD∞ b (hol(X)) (M, N) = Hom (IM , IN ). In this way one obtains an A∞ -model for the derived category of the category of holonomic modules. b The discussion above shows certain similarity between D∞ (hol(X)) and naive F (X). At the same time their deformation theories induce different products on the cohomology of X. In case of the Fukaya category it is the quantum product, while in case of hol(X) it is the usual cup product. The reader will also notice that Maslov index is not visible in the case of hol(X). On the other hand we will explain in the next section that: a) The algebras EndD∞ b (hol(X)) ((L, ρ)) and EndF naive (X) ((L, ρ)) are A∞ equivalent. b) If L and L′ are Hamiltonian isotopic, then we can “twist” the space ′ ′ HomD∞ b (hol(X)) ((L, ρ), (L , ρ )) in such a way that it becomes quasi-isomorphic to the corresponding complex of morphisms in F naive (X). 5 Comparison of the categories Let (L, ρ) be a pair as before, i.e. L is a Lagrangian submanifold of X and ρ is a local system on L. Let us denote by E(L,ρ) the correspodning object of F (X), and by V(L,ρ) the corresponding object of hol(X). We assume that m0 = 0 in the A∞ -algebra A(L, ρ). This means that in fact we are dealing with the category F naive (X). We also assume the conditions imposed on L in [FOOO]. This allows us to make necessary choices without further explanations. In particular L is relatively spin in the sense of [FOOO], so the moduli spaces of pseudo-holomorphic discs are orientable. Taking the Hochschild complex of A(L, ρ) we can construct the formal pointed dgmanifold ME(L,ρ) of deformations of E(L,ρ) . Similarly we can start with the Lie algebra HomD∞ b (hol(X)) (V(L,ρ) , V(L,ρ) ) and construct the formal pointed dgmanifold MV(L,ρ) of deformations of V(L,ρ) . b Let us imagine that both F naive (X) and D∞ (hol(X)) are “sheaves of A∞ -categories” on the “moduli space of objects”. Let us also imagine that there is a well-defined moduli space LagrX of Lagrangian submanifolds of X. Then we should have a natural projection π : MX → LagrX . If L is a Lagrangian submanifold, and [L] ∈ LagrX the corresponding point of the 13 moduli space then the fiber π −1 ([L]) consists of local systems supported on L (or on any Lagrangian submanifold representing the same equivalence class in the moduli space). We would like to compare F naive (X) and hol(X) in a “small neighborhood of [(L, ρ)]”. From the categorical point of view we have two A∞ -categories A and B with the same “space” of objects, and such that for any object X the A∞ -algebras EndA (X) and EndB (X) are equivalent. We would like to find a functor Φ : A → A such that changing morphisms in A to Homnew A (X, Y ) = HomA (X, Φ(Y )) one gets a new A∞ -category equivalent to B. 5.1 Main conjecture We will denote by hol(L) the full subcategory of hol(X) consisting of holonomic modules with the given support L. For simplicity we will assume the rationality condition imposed on the symplectic form ω. Hence the Fukaya category is defined over C((q)) (this is not a serious restriction because one can consider deformation quantization over any pro-nilpotent algebra, in particular Cε ). Our main idea can be explained such as follows. Having in mind the intuitive picture of the previous subsection we consider both F naive (X) and b D∞ (hol(X)) in a small neighborhood of a given [L] ∈ LagrX . For a pair b (hol(L2 )) → L1 , L2 sufficiently close to L we would like to find a functor ΦL1 ,L2 : D∞ b b D∞ (hol(L1 )), such that Hom(ρ1 , ΦL1 ,L2 (ρ2 )) (morphism as objects of D∞ (hol(L1 ))) is quasi-isomorphic to HomF naive (X) ((L1 , ρ1 ), (L2 , ρ2 )). Such a functor should be represented by a bimodule. Let us describe all this more precisely. Let Mi = V(Li ,ρi ) ∈ hol(Li ), i = 1, 2. We expect there exists a Lagrangian submanifold Λ12 = Λ(L1 , L2 ) ⊂ X × X and K(L1 , L2 ) = KΛ12 ∈ hol(X × X) such that: 1) If L1 and L2 have a non-empty intersection then Λ12 ◦ L1 = L2 . Here Λ ◦ L = π2 (π1−1 (L) ∩ Λ), where πi : X × X → X, i = 1, 2 are the natural projections. In particular we assume that the restrictions of πi , i = 1, 2 to Λ are coverings. 2) HomD∞ b (hol(X)) (M1 , K(L1 , L2 ) ◦ M2 ) ≃ HomF naive (X) ((L1 , ρ1 ), (L2 , ρ2 )), where ≃ means quasi-isomorphism of complexes, and we consider both categories over the field C((q)) (i.e. q = t in the case of hol(X)). The composition ◦ for modules is given by the formula K ◦ M = π2∗ (K ⊗ π1∗ (M)). We denote by Homnew (M1 , M2 ) the left hand side of 2). 3) For a generic sequence of Lagrangian submanifolds L1 , L2 , ..., Ln , n ≥ 2 14 and holonomic modules M1 , ..., Mn such that Mi = V(Li ,ρi ) for all i, we expect to have an isomorphism of AX n -modules K(L1 , L2 ) ◦ K(L2 , L3 ) ◦ ... ◦ K(Ln−1 , Ln ) → K(L1 , Ln ). Such an isomorphism defines a linear map mnew : ⊗1≤i≤n−1 Homnew (Mi , Mi+1 ) → Homnew (M1 , Mn ). n We expect the above data satisfy the following Conjecture 1 (i) Higher compositions mnew n , n ≥ 1 give rise to a structure b of an A∞ -category on D∞ (hol(X)). (ii) This category is A∞ -equivalent to F naive (X, ω) (with q = t). 5.2 The conjecture in the case of cotangent bundle Let us fix a Lagrangian submanifold L ⊂ X, and consider only those L′ which are “very close” to L. More precisely we assume that they are not only close to L but also Hamiltonian isotopic to L. We want to “restrict” F naive (X) to this “neighborhood of L”. This means that we consider an A∞ -subcategory with the objects taken from the above-mentioned subset, and morphisms b same as in F naive (X). We do the same thing with hol(X) and D∞ (hol(X)). We would like to compare these categories in the case when X = T ∗ Y is the cotangent bundle with the standard symplectic structure (notice that a neighborhood of a Lagrangian submanifold L can be identified by a symplectomorphism with a neighborhood of the zero section in T ∗ L). We are going to consider Lagrangian submanifolds of the type Li = {(x, dfi (x))|x ∈ Y }. Let ρi be local systems on Li . For a pair of such Lagrangian submanifolds we have a symplectomorphism φ : X → X such that (x, ξ) 7→ (x, ξ + df2 (x) − df1 (x)). Clearly it maps isomorphically L1 into L2 . Let us define Λ = Λ12 as graph(φ) ⊂ X ×X. The corresponding bimodule KΛ is the quotient of AX ⊠ Aop X by the left ideal generated by the relation 1 a ⊗ 1 = 1 ⊗ e t ad(f2 −f1 ) (a), a ∈ AX . Here ad(a)(b) = ab − ba (clearly AX contains CY∞ as a subalgebra, so the ideal is well-defined). Notice that exp 1t (ad(f2 − f1 ))exp 1t (ad(f3 − f2 ))...exp 1t (ad(fn − fn−1 )) = exp 1t (ad(fn − f1 )). Hence we have an isomorphism K(L1 , L2 )◦K(L2 , L3 )◦...◦K(Ln−1 , Ln ) → K(L1 , Ln ). 15 In order to check the conjecture we may assume that f1 = 0. Then we observe that • ∗ HomD∞ b (hol(X)) (ρ1 , K(L1 , L2 ) ◦ ρ2 ) = Ω (Y, ρ1 ⊗ ρ2 ) where the RHS is the complex of de Rham forms with values in the local system. Let ∇i , i = 1, 2 denotes the flat connection on ρi , i = 1, 2. Then the differential is given by ∇∗1 ⊗1+1⊗∇2 +df2 ∧(·). The latter complex is equivalent to the standard de Rham complex (without df2 ) if one twists the sections by exp( 1t f2 ), i.e. s 7→ s exp( 1t f2 ). Then according to [KoSo1], Section 4 the resulting complex is quasi-isomorphic to HomF naive (X) ((L1 , ρ1 ), (L2 , ρ2 )). We remark that in the notation of [KoSo1] one has q = exp(− 1ε ). Remark 4 The case of general symplectic manifold does not follow automatically from the results of this section. Indeed, in order to define morphisms in the Fukaya category for Lagrangian submanifolds in a small neighborhood of a given L one has to consider pseudo-holomorphic discs which do not belong entirely to the neighborhood (we are restricted by the boundary conditions only). Nevertheless, if our main conjecture is true, one can find a family of kernels K(L1 , L2 ) which takes care about such discs. 5.3 Complex structure on the moduli space of holonomic modules We observe that the tangent space to the moduli space of deformations of a module M ∈ hol(X), supp(M) = L is isomorphic to HomD∞ b (hol(X)) (M, M) (derived deformations of Homhol(X) (IdX , IdX )). There is a natural embedding HomD∞ b (hol(L)) (M, M) → HomD b (hol(X)) (M, M), corresponding to the ∞ deformations with the fixed support L. On the other hand there is a natu• • ral projection HomD∞ b (hol(X)) (M, M) → Ω (L), where Ω (L) denotes the de Rham complex of L. Indeed, a deformation of the module M induce the deformation of the support of M. The latter are controlled by differential forms on the support. We can perform computations in the derived categories. Then we have an exact sequence of the tangent spaces to the formal moduli spaces of deformations: • Ext•hol(L) (M, M) → Ext•hol(X) (M, M) → HDR (L). 16 Suppose that M is a simple module. Then RHomhol(L) (M, M) ≃ RΓ(L, RHom(M, M)) ≃ RΓ(L, CL ) ≃ Ω• (L). Taking first cohomology (this corresponds to “classical” tangent space) we obtain in this case an exact sequence 1 1 HDR (L) → Ext1hol(X) (M, M) → HDR (L). This means that the tangent space to the “classical” deformations of M inside of hol(X) is twice as big as the tangent space to the “classical” deformations of L inside of LagrX . If X is a Calabi-Yau manifold then one hopes to obtain a complex structure on the moduli space of formal deformations of a simple holonomic module M. We expect it is isomorphic to a subvariety in the dual Calabi-Yau manifold. In order to describe the complex structure explicitely we need to identify the tangent space TL (LagrX ) with the tangent space TM (hol(L)). It is sufficient to describe the lifting of paths from LagrX to hol(L). Given a path L(t) ⊂ LagrX such that L(0) = L, we define a path M(t) ⊂ hol(L), M(0) = M in the following way: M(t) = M ⊗ ρ(t). Here ρ(t) is the restriction to L(t) of the unitary bundle over X with a connection ∇ such that curv(∇) = ωX (it is often called the pre-quantum line bundle). Restriction of ∇ to L(t) is a flat connection. 6 Conclusion We have suggested the way to compare the Fukaya category with the category of holonomic modules over the quantized algebra of smooth functions. The idea is for a pair of Lagrangian submanifolds L1 , L2 to find a kernel K(L1 , L2 ) which transforms local systems (or more general AX -modules) supported on L2 to local systems supported on L1 . We have conjectured that it is possible to make these choices in such a way that the counting of instantons (i.e. higher compositions in the Fukaya category) can be replaced by pure algebraic operation of taking homomorphisms between local systems having the same Lagrangian support. We have checked the conjecture in the simplest case. It would be interesting to check it in other cases as well as “globalize” this picture, finding kernels K(L1 , L2 ) for Lagrangian submanifolds which are not close to each other. 17 References [AP] D. Arinkin, A. 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