Fukaya category (of a compact symplectic manifold) in the approach presented here will be in fact an $A_{\infty}$-pre-category. Our definition is not given in the maximal generality, but it will be sufficient for the main application to abelian varieties. For more elaborated definitions see [Fu1], [Ko].

Let $(V,\omega)$ be a compact symplectic manifold of dimension $2n$, such that $c_{1}(T_{V})=0\in H^{2}(V,{{\bf Z}})$. The Fukaya category (with the trivial $B$-field) associated with $(V,\omega)$ depends on some additional data, which we are going to describe below.

We fix an almost complex structure $J$ compatible with $\omega$ and a smooth everywhere non-vanishing differential form $\Omega$, which is $(n,0)$-form with respect to $J$. Let $L$ be an oriented Lagrangian submanifold. Then one has a map $Arg_{L}:=Arg_{\Omega_{|L}}:L\to{{\bf R}}/2\pi{{\bf Z}}$, where $Arg_{\Omega_{|L}}(x)$ is the argument of the non-zero complex number $\Omega(e_{1}\wedge...\wedge e_{n})$, and $e_{1},...,e_{n}$ is an oriented basis of $T_{x}L,x\in L$.

We will sometimes denote the Fukaya category by $F(V,\omega)$, or simply by $F(V)$. We will also often omit from the notation the lifted argument function, thus denoting an object simply by $(L,\rho)$.

Let ${{\bf C}}_{\varepsilon}$ be the field consisting of formal series $f=\sum_{i\geq 0}c_{i}e^{-\lambda_{i}/\varepsilon}$, such that $c_{i}\in{{\bf C}},\lambda_{i}\in{{\bf R}},\lambda_{0}<\lambda_{1}<...,\lambda_{i}\to+\infty$. In the case when $[\omega]\in H^{2}(V,{{\bf Z}})$, one can in fact work over the field ${{\bf C}}((q))$, where $q=exp(-{1\over{\varepsilon}})$. In general we equip ${{\bf C}}_{\varepsilon}$ with the adic topology: a fundamental system of neighborhoods of zero consists of sets $U_{x}=\{f=\sum_{i\geq 0}c_{i}e^{-\lambda_{i}/\varepsilon}|\lambda_{i}\geq x,i\geq 0\},x\in{{\bf R}}$.

Thus morphisms form a finite-dimensional vector space over the field ${{\bf C}}_{\varepsilon}$. There is a ${\bf Z}$-grading of the space of morphisms given in terms of Maslov index $deg:L_{1}\cap L_{2}\to{{\bf Z}}$ (see [Fu2], [Ko], [Se]).

Now we are going to describe the $A_{\infty}$-structure. It is defined by means of a collection of maps (higher compositions) of graded vector spaces $m_{k}^{F(V)}:\otimes_{0\leq i\leq k-1}Hom_{F(V)}((L_{i},\rho_{i}),(L_{i+1},\rho_{i+1}))\to Hom_{F(V)}((L_{0},\rho_{0}),(L_{k},\rho_{k}))[2-k]$, where $k\geq 1$ and the sequence $(L_{0},...,L_{k})$ corresponds to a transversal sequence of objects (the latter notion will be defined below).

In the case, when all local systems are trivial of rank one, the map $m_{k}$ is defined such as follows. Let $D$ be a standard disc $D=\{z\in{{\bf C}}|\,|z|\leq 1\}$. Let us fix a sequence $(L_{0},...,L_{k})$ of supports of objects with pairwise transversal intersections, intersection points $x_{i}\in L_{i}\cap L_{i+1},0\leq i\leq k-1$, $x_{k}\in L_{0}\cap L_{k}$, and $\beta\in\pi_{2}^{free}(V,\cup_{0\leq i\leq k}L_{i})$. We denote by ${\cal M}(L_{0},...,L_{k};x_{0},...,x_{k};\beta)$ the set of collections $(y_{0},...,y_{k};\psi)$, where $y_{i},0\leq i\leq k$ are cyclically ordered pairwise distinct points on the boundary $\partial D$, and $\psi:D\to(V,J)$ a pseudo-holomorphic map such that $\psi(y_{i})=x_{i},\psi(\overline{y_{i}y_{i+1}})\subset L_{i},0\leq i\leq k,y_{0}=y_{k}$, $[\phi]=\beta$. Here $\overline{y_{i}y_{i+1}}$ denotes the arc between $y_{i}$ and $y_{i+1}$. There is a natural action of $PSL(2,{{\bf R}})$ on ${\cal M}(L_{0},...,L_{k};x_{0},...,x_{k};\beta)$ arising from the holomorphic action on $D$ by fractional linear transformations. The action is free except of the case $k=1,x_{0}=x_{1},\beta=0$, which is not relevant for our purposes.

Let $x_{i}\in L_{i}\cap L_{i+1},0\leq i\leq k-1,x_{k}\in L_{0}\cap L_{k}$ satisfy the condition $deg\,x_{k}=\sum_{0\leq i\leq k-1}deg\,x_{i}+2-k$. Then the matrix element $(m_{k}(x_{0},x_{1},...,x_{k-1}),x_{k})$ is given by the formula $(m_{k}(x_{0},x_{1},...,x_{k-1}),x_{k})=\sum\pm q^{({\beta},[\omega])}$, where sum is taken over all $PSL(2,{{\bf R}})$-orbits of points in ${\cal M}(L_{0},...,L_{k};x_{0},...,x_{k};\beta)$. Signs are derived from orientations of certain cycles in the moduli space ${\cal M}={\cal M}(L_{0},...,L_{k};x_{0},...,x_{k};\beta)/PSL(2,{{\bf R}})$. We will comment on them below (see [Fu1], [Ko] for more details). In the case of non-trivial local systems there is an additional factor for each summand. It corresponds to the holonomies of local system along the arcs.

Now we will describe the transversality condition. Assume that we are given a sequence of objects $(L_{i},\rho_{i}),0\leq i\leq k$ of the Fukaya category. We say that they are transversal if the following conditions hold:

1) There are only pairwise intersections $L_{i}\cap L_{j}$, and they are transversal.

2) For any subsequence $(L_{i_{0}},...,L_{i_{m}}),\,m\geq 1,\,i_{0}<i_{1}<...<i_{m}$, any choice of intersection points $x_{i_{m}}\in L_{i_{0}}\cap L_{i_{m}}$, $x_{i_{p}}\in L_{i_{p}}\cap L_{i_{p+1}},0\leq p\leq m-1$ such that $deg\,x_{i_{m}}-(\sum_{0\leq i\leq m-1}deg\,x_{i_{p}}+2-m)=0$, and any $\beta\in\pi_{2}^{free}(V,\cup_{0\leq p\leq m}L_{i_{p}})$, the corresponding component of the moduli space
${\cal M}(L_{i_{0}},...,L_{i_{m}};x_{i_{0}},...,x_{i_{m}};\beta)/PSL(2,{{\bf R}})$ contains only smooth points, and is zero-dimensional.

3) If $deg\,x_{i_{m}}-(\sum_{0\leq i\leq m-1}deg\,x_{i_{p}}+2-m)<0$ then the corresponding component is empty.

Let us comment on these conditions. The first one is needed to define morphisms. The quotient set ${\cal M}={\cal M}(L_{i_{0}},...,L_{i_{m}};x_{i_{0}},...,x_{i_{m}};\beta)/PSL(2,{{\bf R}})$ which appears in the second condition locally can be identified with the space of solutions of a non-linear elliptic problem. For the linearized problem the corresponding Fredholm operator has index $deg\,x_{i_{m}}-(\sum_{0\leq i\leq m-1}deg\,x_{i_{p}}+2-m)$. We define smooth points ${\cal M}^{sm}$ of ${\cal M}$ as such points where the cokernel of the Fredholm operator is trivial. Then ${\cal M}^{sm}$ is a smooth manifold of the dimension equal to the index. Moreover, one checks that the spaces ${\cal M}^{sm}$ carry natural orientations given by the determinants of the corresponding Fredholm operators. It follows that in the zero-dimensional case what we get is a set of points with multiplicities $\pm 1$ (in particular, the multiplicities are integer numbers). Multiple covers and stable maps which appear in the definition of Gromov-Witten invariants and produce non-trivial denominators, do not appear in our framework for the Fukaya category. Therefore one can define the Fukaya category over the ring ${{\bf Z}}_{\varepsilon}$ (the integral version of ${{\bf C}}_{\varepsilon}$). The number of points counted with signs gives a tensor coefficient of $m_{k}$.

Composition maps satisfy a system of quadratic equations, thus making $F(V,\omega)$ into a non-unital $A_{\infty}$-pre-category. One can check that it is in fact an $A_{\infty}$-pre-category. Proof of the extension property is based on the following result of Fukaya (see [Fu2], [Se]).

For example, a sequence consisting of one object $(L,\rho)$ can be extended to a transversal sequence $((L_{t_{1}},\rho_{t_{1}}),(L,\rho))$. Similarly, one can extend any finite set of transversal sequences.

It is easy to see that the set of connected components of the space of pairs $(J,\Omega)$ (equipped with the natural topology) is a principal homogeneous space over the lattice $H^{1}(V,{{\bf Z}})$. Namely, $f:V\to U(1)$ acts on $(J,\Omega)$ such as follows: $(J,\Omega)\mapsto(J,f\Omega)$. The following theorem can be derived from [Fu2].

Therefore the equivalence class of the Fukaya category depends on the connected component of the space of pairs.

Let $(Y,g_{Y},\nabla)$ be an AK-manifold with integral affine structure. The covariant lattice is denoted by $T_{Y}^{{\bf Z}}$, as before. From now on we will assume that $Y$ is compact. This is a severe restriction. It was proven in [CY] that in this case a finite cover of space $Y$ is a torus with the standard affine structure. It appears in the collapse of complex abelian varieties.

The manifold $X^{\vee}=T_{Y}^{\ast}/(T_{Y}^{{\bf Z}})^{\vee}$ is the total space of the torus bundle $p^{\vee}:X^{\vee}\to Y$. It carries a natural symplectic form $\omega=\omega_{X^{\vee}}$ induced from the standard one on $T^{\ast}Y$. We endow $X^{\vee}$ with a 1-parameter family of complex structures $J_{\eta},\eta\to 0$ compatible with $\omega$. Indeed, the manifold $X_{\eta}^{\vee}:=T_{Y}^{\ast}/\eta(T_{Y}^{{\bf Z}})^{\vee}$ carries a canonical complex structure described before. We identify $X^{\vee}$ and $X_{\eta}^{\vee}$ by the map $(y,v)\mapsto(y,\eta v)$, where $y\in Y,v\in T^{\ast}_{Y,y}$. Using this identification, we pull back to $X^{\vee}$ the complex structure and the metric. The fibers of $p^{\vee}:X^{\vee}\to Y$ are flat Lagrangian tori for all values of $\eta$.

We define on $(X^{\vee},J_{\eta})$ a nowhere vanishing $(n,0)$-form $\Omega_{\eta}$ such as follows. Let us fix an oriented orthonormal basis $e_{1},...,e_{n}$ in $T_{Y,y}^{\ast},y\in Y$. We define $\Omega_{\eta}$ as the $n$-form on $X^{\vee}$, which is invariant with respect to the $T_{Y,y}^{\ast}/(T_{Y,y}^{{\bf Z}})^{\vee}$-action, and is equal to $\bigwedge_{1\leq j\leq n}((p^{\vee})^{\ast}e_{j}+\sqrt{-1}J_{\eta}(p^{\vee})^{\ast}e_{j})$.

Let $L$ be a compact oriented Lagrangian submanifold of $X^{\vee}$ such that $p^{\vee}_{|L}$ is an unramified covering, and the orientation of $L$ is induced from the orientation of $Y$. We claim that there is a canonical choice $\widetilde{Arg}_{L}^{can}:L\to{{\bf R}}$ for the function $\widetilde{Arg}_{L}:L\to{{\bf R}}$. Indeed, for any point $x\in X^{\vee}$ the space of Lagrangian subspaces in $T_{X^{\vee},x}$, which are transversal to the vertical tangent space $T_{x}^{vert}=Ker(p^{\vee})_{\ast}$ is contractible. Let us consider the space ${\cal L}$ of pairs $(x,l)$ such that $x\in X^{\vee}$ and $l\subset T_{X^{\vee},x}$ is a Lagrangian subspace, which is transversal to $T_{x}^{vert}$, and endowed with the orientation induced from $Y$. Then the function $(x,l)\mapsto Arg_{(\Omega_{\eta})_{|l}}(x)\in{{\bf R}}/2\pi{{\bf Z}}$ admits a unique continuous lifting $\widetilde{Arg}:{\cal L}\to{{\bf R}}$, vanishing at $(x,J_{\eta}(T_{x}^{vert})),x\in X^{\vee}$. Restricting this function to $L$ we obtain $\widetilde{Arg}_{L}^{can}$.

We will denote by $F^{\eta}(X^{\vee})$ the Fukaya category $F(X^{\vee},\omega,J_{\eta},\Omega_{\eta})$, and by $F_{unram}^{\eta}(X^{\vee})$ its full $A_{\infty}$-pre-subcategory with objects $(L,\rho,\widetilde{Arg}_{L}^{can})$ such that $L$ is a compact Lagrangian submanifold with the orientation induced from $Y$, $p^{\vee}_{|L}$ is an unramified covering, and $\widetilde{Arg}_{L}^{can}$ was described above. To simplify the notations we will denote objects of these categories by $(L,\rho)$.

It follows from the results of [FuO] that there exists a limit of the family of $A_{\infty}$-pre-categories $F_{unram}^{\eta}(X^{\vee})$, $\eta\to 0$ in the following sense. Objects and morphisms of $F_{unram}^{\eta}(X^{\vee})$ do not depend on $\eta$ and remain the same in the limit. The compositions $m_{k}^{F_{unram}^{\eta}(X^{\vee})}$ have limits as $\eta\to 0$ in the adic topology of ${{\bf C}}_{\varepsilon}$. They will be explicitly described below.

The following result can be derived from [FuO].

We will denote this $A_{\infty}$-pre-category by $FO(X^{\vee})$ and call it the Fukaya-Oh category of $X^{\vee}$ (or degenerate Fukaya category of $X^{\vee}$).

As we said before, the objects and morphisms for $FO(X^{\vee})$ are the same as for $F_{unram}^{\eta}(X^{\vee})$. In order to define the composition map

one uses the standard formulas, but the sum runs over certain two-dimensional surfaces in $X^{\vee}$ described below. For a sequence $((L_{0},\rho_{0}),...,(L_{k},\rho_{k})),k\geq 1$ of objects in $FO(X^{\vee})$ we consider immersed two-dimensional surfaces $S\to X^{\vee}$ such that:

a) Boundary of $S$ belongs to $L_{0}\cup...\cup L_{k}$.

b) $S=(\cup_{\alpha}T_{\alpha})\cup(\cup_{\beta}S_{\beta})$ where and $T_{\alpha}$ are geodesic triangles in fibers of $p^{\vee}$, hence they are projected to points in $Y$.

c) Each $S_{\beta}$ is a union of 1-parameter families of geodesic intervals contained in fibers of $p^{\vee}$ (i.e. a “strip”). Moreover, $p^{\vee}_{|S_{\beta}}:S_{\beta}\to Y$ is a fibration over a connected interval $I_{\beta}:=p^{\vee}(S_{\beta})$ immersed in $Y$. Fibers of $S_{\beta}$ over the interior points of $I_{\beta}$ are geodesic intervals of strictly positive length. Fibers of $S_{\beta}$ over the boundary points of $I_{\beta}$ are either edges of triangles $T_{\alpha}$ or intersection points $x_{i}\in L_{i}\cap L_{i+1},0\leq i\leq k-1,\,x_{k}\in L_{0}\cap L_{k}$.

d) Intervals $I_{\beta}$ are edges of an immersed planar trivalent tree $\Gamma\subset Y$. Points $p^{\vee}(T_{\alpha})$ are internal vertices of $\Gamma$. Tail vertices of $\Gamma$ are projections of the intersection points $x_{0},\dots,x_{k}$.

e) Let $r:T^{\ast}_{Y}\to X^{\vee}$ be the natural fiberwise universal covering. If the Lagrangian manifolds $r^{-1}(L_{i}),i=0,...,k$ are locally given by differentials of smooth functions $f_{i},i=0,...,k$ on $Y$, then the edges of $\Gamma$ must be gradient lines of $f_{i}-f_{j}$. Intersection points of $r^{-1}(L_{i})$ and $r^{-1}(L_{j})$ correspond to critical points of $f_{i}-f_{j}$.

We depict a typical surface below:

The projection of surface $S$ to $Y$ is a gradient tree, with tail vertices being critical points of $f_{i}-f_{i+1}$ or of $f_{0}-f_{k}$, and edges $p^{\vee}(S_{\beta})$ being the gradient lines of functions $f_{i}-f_{j}$, where $i=i(\beta),j=j(\beta),i<j$. The triangles are mapped into the internal vertices of the tree. Here is the picture of $\Gamma=p^{\vee}(S)$ for surface $S$ as above:

Compositions $m_{k}=m_{k}^{FO(X^{\vee},\omega)}$ are given by the standard formulas, but now we are counting surfaces $S$ described in a)-d). The weight $q^{\langle[S],[\omega]\rangle}$ can be written as $exp(-{1\over{\varepsilon}}\sum_{\beta}var_{p^{\vee}(S_{\beta})}f_{\beta})$, where $f_{\beta}=f_{i(\beta)}-f_{j(\beta)}$, and var is the (positive) variation of the function along the gradient line.

The transversality condition for a sequence of objects of Fukaya-Oh category can be formulated similarly to the case of Fukaya category.

The reader can compare our considerations with those from [FuO]. The fibers of $p^{\vee}$ are “small” tori (of the size $O(\eta))$. The base $Y$ is “large” (of the size of $O(1)$). Hence, the Lagrangian manifolds are close to the zero section of $p^{\vee}$. This is similar to the situation considered in [FuO]. Indeed, in [FuO] the authors study the $A_{\infty}$-subcategory of $F(T^{\ast}_{Y})$ (where $Y$ is an arbitrary smooth compact manifold), with the objects $(L,\rho)$ such that $L=\eta\,graph(df)$, $f:Y\to{{\bf R}}$ is a smooth function. In other words, they considered Lagrangian sections of the natural projection $T^{\ast}_{Y}\to Y$, which are close to the zero section. When $\eta\to 0$, pseudo-holomorphic discs get “stretched” along the fibers of $p^{\vee}$. Thus they look like the surfaces $S$ described above. Then the higher compositions of the Fukaya category “approach” the compositions $m_{k}^{FO(X^{\vee},\omega)}$. This was proved in [FuO] in the case when $X^{\vee}$ was replaced by $T^{\ast}_{Y}$. Considerations from [FuO] apply in our case as well.