In the case of complex numbers the mirror symmetry functor assigns a holomorphic vector bundle $F(L,\rho)$ on $X=X_{\varepsilon}$ to a pair $(L,\rho)$, where $L\subset X^{\vee}$ is a Lagrangian submanifold, such that the projection $p^{\vee}_{|L}:L\to Y$ is an unramified covering, and $\rho$ is a local system on $L$. If $L$ is a section of $p^{\vee}$, and $rank(\rho)=1$, then $E=F(L,\rho)$ is a line bundle. In general, $E$ can be locally represented as a sum $E\simeq\oplus_{\alpha\in A}E_{\alpha}$ where $A$ is the set of leaves (i.e. connected components) of the covering $L\to Y$, and $E_{\alpha}$ is a holomorphic vector bundle of the rank equal to the rank of $\rho$ at the leaf $\alpha$.

The following explicit construction of the mirror symmetry functor on objects is not new, see e.g. [AP]. We start with the remark that there is a canonical $U(1)$-bundle on $X\times_{Y}X^{\vee}$ (Poincaré line bundle). It will be denoted by ${P}$. It admits a canonical connection, which will be described below . Let us fix $y\in Y$. Then $p^{-1}(y)\simeq T_{Y,y}/\varepsilon T_{Y,y}^{\bf Z}$ and $(p^{\vee})^{-1}(y)\simeq T_{Y,y}^{\ast}/(T_{Y,y}^{\bf Z})^{\vee}$. We identify torus $(p^{\vee})^{-1}(y)$ with the moduli space of $U(1)$-local systems on the torus $p^{-1}(y)$ trivialized over a point $0\in p^{-1}(y)$. We define $U(1)$-bundle $P$ to be the tautological bundle on $X\times_{Y}X^{\vee}$ corresponding to this description.

In order to describe the connection on $P$ let us consider the fiberwise universal coverings $r:T_{Y}\to T_{Y}/T_{Y}^{\bf Z}$ and $r^{\vee}:T_{Y}^{\ast}\to T_{Y}^{\ast}/(T_{Y}^{\bf Z})^{\vee}$. Then the pullback $\bar{P}$ of $P$ to $T_{Y}\times_{Y}T_{Y}^{\ast}$ is canonically trivialized. Thus we can work in coordinates. Let $y=(y_{1},...,y_{n})$ be coordinates on $Y$, $x=(x_{1},...,x_{n})$ and $x^{\vee}=(x_{1}^{\vee},...,x_{n}^{\vee})$ be coordinates on the fibers of $T_{Y}\to Y$ and $T_{Y}^{\ast}\to Y$ respectively. Deck transformations $x_{j}\mapsto x_{j}+\varepsilon n_{j},\,n_{j}\in{\bf Z}$ act on $\bar{P}$ preserving the trivialization, and transformations $x_{j}^{\vee}\mapsto x_{j}^{\vee}+n_{j}^{\vee},\,n_{j}^{\vee}\in{\bf Z}$ act on $\bar{P}$ by the multiplication by $exp(2\pi i/\varepsilon\sum_{j}n_{j}^{\vee}x_{j})$.

Let $\nabla_{0}$ be the trivial connection on $\bar{P}$. We consider the connection $\bar{\nabla}$ on $\bar{P}$ which is given by the following formula

Proof. Obviously, connection $\bar{\nabla}$ does not change under the transformation $x_{j}\mapsto x_{j}+\varepsilon n_{j},n_{j}\in{\bf Z}$. The transformation $x_{j}^{\vee}\mapsto x_{j}^{\vee}+n_{j}^{\vee},n_{j}^{\vee}\in{\bf Z}$ together with the gauge transformation of $\bar{\nabla}$ by $h=exp(2\pi i/\varepsilon\sum_{j}n_{j}^{\vee}x_{j})$ also preserves $\bar{\nabla}$. This proves the Lemma. $\blacksquare$

Let $(L,\rho)$ be as above. The mirror symmetry functor assigns to it a holomorphic vector bundle $E=F(L,\rho)$ such that (in coordinates) its fiber over a point $(y,x)$ is given by the formula $E(y,x)=\oplus_{\{x^{\vee}\in L,p^{\vee}(x^{\vee})=y\}}\rho(x^{\vee})\otimes P(x,x^{\vee})$. This vector bundle carries the induced connection $\nabla_{E}$. In the case of unitary $\rho$ the bundle $E$ carries also a natural hermitean metric.

Proof. It follows from the fact that $L$ is Lagrangian. Indeed, let us lift $L$ to $T_{Y}^{\ast}$. Then locally in a neighborhood of a connected component of $L$, one can find a smooth real function $f=f(y)$ such that $L=df$. We can write the local equation for $L$: $x_{j}^{\vee}=\partial f/\partial y_{j},1\leq j\leq n$. The connection $\nabla_{E}$ can be locally written as $\nabla_{E,0}+id_{E}\otimes(2\pi i/\varepsilon\sum_{j}\partial f/\partial y_{j}dx_{j})$, where $\nabla_{E,0}$ is the trivial flat connection on the vector bundle $E$. Since the holomorphic coordinates on $T_{Y}$ are given by $z_{j}=y_{j}+ix_{j},i=\sqrt{-1}$, one sees that the $(0,2)$-part of the curvature is equal to $curv(\nabla_{E})^{(0,2)}=const\times(\sum_{j,k}\partial^{2}f/\partial y_{j}\partial y_{k}d\bar{z_{j}}d\bar{z_{k}})=0$. The Proposition is proved. $\blacksquare$

We consider the space of Dolbeault differential forms with values in the vector bundle $Hom(E_{1},E_{2})$ as a dg-algebra with respect to the $\bar{\partial}$-differential. In this way one gets a structure of $A_{\infty}$-category (in fact a dg-category) on the derived category of coherent sheaves on $X$. One can show that this $A_{\infty}$-structure is equivalent to the one mentioned in the main text.