The purpose of this section is to describe the framework in which the results concerning $A_{\infty}$-categories will be formulated. We would like to make few comments even before recalling a definition of the Fukaya category. There are two main problems with the definition. First, morphisms can be defined only for transversal Lagrangian submanifolds (in particular, the identity morphism is never defined). Second, since there are pseudo-holomorphic discs with the boundary on a given Lagrangian submanifold, one has to add a composition $m_{0}$ to the set of compositions $m_{n},n\geq 1$. As a result, the spaces of morphisms are not complexes: $m_{1}^{2}\neq 0$. On the other hand, the derived category of coherent sheaves arises from an $A_{\infty}$-category without $m_{0}$ and with the condition $m_{1}^{2}=0$. Hence one should explain in which sense two $A_{\infty}$-categories in question are equivalent.

The above-mentioned problems can be resolved by an appropriate generalization of the notion of $A_{\infty}$-category. This generalization involves numerous preparations and will be given elsewhere (see [KoS]). On the other hand, the problem with $m_{0}$ does not appear in the case of abelian varieties, which is the main application of the approach offered in this paper. Hence, for the purposes of present paper it is sufficient to work with $A_{\infty}$-pre-categories (or $A_{\infty}$-categories with transversal structure, cf. [P1]). This gives a partial solution to the transversality problem, and provides a solution to the problem with the identity morphisms.

Using $A_{\infty}$-pre-categories we formulate and prove a variant of the homological mirror symmetry conjecture. It can be applied to the case of abelian varieties. In particular, one can obtain certain formulas for Massey products for abelian varieties in terms of partial theta-sums similar to those considered in [P1].

Let $A=\oplus_{i\in Z}A^{i}$ be a ${\bf Z}$-graded module over a ${\bf Z}$-graded commutative associative algebra $k$. As usual, we will denote by $A[n]$ the graded $k$-module such that $(A[n])^{i}=A^{i+n}$ for all $i$.

The codifferential $d$ is by definition a coderivation, such that $d^{2}=0$. It is uniquely determined by its “Taylor coefficients” $m_{n}:A^{\otimes n}\to A[2-n],n\geq 1$. The condition $d^{2}=0$ can be rewritten as a sequence of quadratic equations

where $a_{m}\in A$, and $\epsilon(l,j)=(-1)^{j\sum_{0\leq s\leq l-1}deg(a_{s})}$. In particular, $m_{1}^{2}=0$.

A morphism $f$ of non-unital $A_{\infty}$-algebras is determined by its “Taylor coefficients” $f_{n}:V^{\otimes n}\to W[1-n],n\geq 1$ satisfying the system of equations

$\sum_{1\leq l_{1}<...,<l_{i}=n}\pm m_{i}^{W}(f_{l_{1}}(a_{1},...,a_{l_{1}}),$
$f_{l_{2}-l_{1}}(a_{l_{1}+1},...,a_{l_{2}}),...,f_{n-l_{i-1}}(a_{n-l_{i-1}+1},...,a_{n}))=$

$\sum_{s+r=n+1}\sum_{1\leq j\leq s}\pm f_{s}(a_{1},...,a_{j-1},m_{r}^{V}(a_{j},...,a_{j+r-1}),a_{j+r},...,a_{n}).$

We leave to the reader as an exercise to write down the formulas for the signs in terms of degrees of $a_{i}$ and $f_{i}$.

The class of objects $Ob({\cal C})$ will be often denoted by ${\cal C}$. We hope it will not lead to a confusion.

We start with the notion of non-unital $A_{\infty}$-pre-category. It allows us to work with ‘‘transversal’’ sequences of objects. ³³ 3 The notion of “transversality” is purely formal in this section. The choice of the name will become clear after concrete applications in the geometric context, see next sections. Then we will introduce the notion of $A_{\infty}$-pre-category. It provides us with a replacement of the identity morphisms. Roughly speaking, we will have the identity morphism up to homotopy.

The reader have noticed that we use the summation only over the increasing pairs of indices $i<j$. It differs from the case of non-unital $A_{\infty}$-pre-categories. The reason is that we do not require the transversality to be a symmetric relation on objects. It is possible that $Hom(X_{0},X_{1})$ exists, but $Hom(X_{1},X_{0})$ does not. In the case when all $Hom^{\prime}s$ are defined, two discussed definitions agree. In particular, a non-unital $A_{\infty}$-category is the same as a non-unital $A_{\infty}$-pre-category such that ${\cal C}_{n}^{tr}={\cal C}^{n}$ for any $n\geq 1$.

There is an important notion of equivalence of $A_{\infty}$-pre-categories (and $A_{\infty}$-categories). We are planning to provide all the details elsewhere (see [KoS]). For the purposes of present paper we will be using the following definition (which is in fact a theorem in the more general framework).

We suggest the language of $A_{\infty}$-pre-categories in order to replace more conventional $A_{\infty}$-categories with strict identity morphisms.

An $A_{\infty}$-category ${\cal C}$ with strict identity morphisms is an $A_{\infty}$-pre-category, because (in the previous notation) we can extend a transversal sequence $S$ to $(X,S,X)$, and set $X_{+}=X_{-}=X$, $f_{\pm}=1_{X}$. Another remark is that if ${\cal C}$ has only one object, it is an $A_{\infty}$-algebra with the strict unit. One can try to develop the deformation theory of such algebras along the lines of [KoS1]. The problem is that the corresponding operad is not free, and the standard theory becomes complicated. We hope that the framework of $A_{\infty}$-pre-categories is appropriate for the purposes of deformation theory of $A_{\infty}$-categories. The following conjecture gives another evidence in favor of such a generalization of $A_{\infty}$-categories.

There is a useful special case of the notion of $A_{\infty}$-pre-category (independently a similar notion was suggested in [Se]).

The extension property is equivalent to the following one: for any object $X_{i}$ there are exist objects $X_{j},j<i$ and $X_{m},m>i$ which are quasi-isomorphic to $X$. Then one can formulate the following version of the previous conjecture.

Having an $A_{\infty}$-category ${\cal C}$ with strict identity morphisms, and countable class of objects, one can construct an infinite sequence of objects $(X_{i})_{i\in{{\bf Z}}}$ such that each objects appears infinitely many times for positive and negative $i$. Then a directed $A_{\infty}$-pre-category ${\cal C}^{\prime}$ is defined by setting $Hom_{{\cal C}^{\prime}}(X_{i},X_{j})=Hom_{{\cal C}}(X_{i},X_{j})$ for $i<j$. All other $Hom^{\prime}s$ are not defined.