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

$$m_{k}:\otimes_{0\leq i\leq k-1}Hom((L_{i},\rho_{i}),(L_{i+1},\rho_{i+1}))\to Hom((L_{0},\rho_{0}),(L_{k},\rho_{k}))[2-k]$$

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.