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.

Let $E_{i}=F(L_{i},\rho_{i}),i=1,2$ be holomorphic vector bundles as above. There is an analog of the dg-category ${\cal C}(Y)$ in the case of complex numbers. We will denote it by ${\cal A}(Y)$. Objects of ${\cal A}(Y)$ are holomorphic vector bundles on $X$ of the type $E=F(L,\rho)$. Morphisms are sections of soft sheaves on $Y$. Namely, we define the sheaf $\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2})$ on $Y$ as the direct image $p_{\ast}(\underline{Hom}_{Dolb}(E_{1},E_{2}))$ (in the self-explained notation). Then $Hom_{{\cal A}(Y)}(E_{1},E_{2})$ are global sections of this sheaf. This sheaf corresponds to the sheaf
$\underline{Hom}_{{\cal C}(Y)}(E_{1},E_{2})$ in the non-archimedean geometry. Let us choose an open affine chart $U\subset Y,U\simeq{{\bf R}}^{n}$. Then $\Gamma(U,\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2}))$ contains a subsheaf of finite Fourier sums with respect to the natural action of the torus
$T^{n}$ on $\Gamma(U\times T^{n},\underline{Hom}_{Dolb}(E_{1},E_{2}))$. Thus we have the sheaf $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ which is an analog of the sheaf $\underline{Hom}_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$ considered in the non-archimedean case. Notice that there exists a natural homomorphism of sheaves $j:p^{\ast}(\underline{\Omega}^{\ast}_{Y})\to\underline{\Omega}^{0,\ast}_{X}$. The image of $j$ consists of Dolbeault forms on $X$ which have coefficients locally constant along fibers of $p$. In local coordinates $j$ is given by the formula $f_{i_{1},...,i_{n}}(y)dy_{i_{1}}\wedge...\wedge dy_{i_{n}}\mapsto f_{i_{1},...,i_{n}}(y)d\overline{z}_{i_{1}}\wedge...\wedge d\overline{z}_{i_{n}}$, where $z_{k}=y_{k}-\sqrt{-1}x_{k},1\leq k\leq n$. It is easy to see that $j$ is compatible with the structure of dg-algebras on de Rham and Dolbeault forms. Thus for a pair of holomorphic vector bundles $E_{1}$ and $E_{2}$ on $X$ we have a canonical structure of dg-module over $\underline{\Omega}^{\ast}_{Y}$ on the sheaf $\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2})$. In the case when $E_{i}=F(L_{i},\rho_{i}),i=1,2$ the subsheaf $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ is also a sheaf of dg-modules over $\underline{\Omega}^{\ast}_{Y}$.

As in the non-archimedean case there is a canonical decomposition of the stalk $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y},y\in Y$ into the direct sum of dg-modules of finite rank over $\underline{\Omega}^{\ast}_{Y}$. Summands are labeled by the homotopy classes $[\gamma]\in P(L_{1},L_{2},y)$ and called sectors. We will denote them by $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}$. Informally, sectors correspond to “Fourier components” of Dolbeault forms in $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y}$ in the direction of torus fibers. Let us describe them more explicitly. For simplicity we will assume that $\rho_{i},i=1,2$ are rank one trivial local systems, and $L_{i},i=1,2$ intersect with each fiber of $p$ at exactly one point. Then near $p^{-1}(y)$ we can write $L_{i}=graph(df_{i})\,(mod(T_{Y}^{{\bf Z}})^{\vee}),i=1,2$, where $f_{i}$ are germs at $y$ of smooth functions on $Y$. ¿From the description of the Poincaré bundle $P$ we deduce that $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y}$ is canonically identified with the space of germs of $\overline{\partial}$-forms near $T^{n}=p^{-1}(y)$, endowed with the twisted differential $\overline{\partial}^{\prime}\alpha=\overline{\partial}\alpha+{i\over{\varepsilon}}\sum\partial f/\partial y_{i}d\overline{z}_{i}\wedge\alpha$, where $f=f_{1}-f_{2}$. Then the sector corresponding to a path $\gamma$ consists of Dolbeault forms $\alpha=\sum_{i_{1},...,i_{n}}exp(i\langle m,x\rangle/\varepsilon)f_{i_{1}...i_{n}}(y)d\overline{z}_{1}\wedge...\wedge d\overline{z}_{n}$. Here vector $m=m(\gamma)$ is the homotopy class of the loop in $T^{n}=p^{-1}(y)$ which is the composition of three paths:

1) the path $[0,1]\to T^{n},t\mapsto t(df_{1})_{y}\,mod\,(T_{Y}^{{\bf Z}})^{\vee}$;

2) the path $\gamma$;

3) the path $[0,1]\to T^{n},t\mapsto(1-t)(df_{2})_{y}\,mod\,(T_{Y}^{{\bf Z}})^{\vee}$.

A choice of sector corresponds to the choice of monomial $z_{1}^{i_{1}}...z_{n}^{i_{n}}$ in the non-archimedean case. Homotopy classes of paths in non-archimedean approach correspond the summands of Fourier series. Locally each sector can be identified with the de Rham complex on $Y$. Namely, to a form $\alpha=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(y)exp(\langle m,x\rangle)d\overline{z}_{i_{1}}\wedge...\wedge d\overline{z}_{i_{n}}$ we assign the form $\alpha_{m}=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(y)exp({1\over{\varepsilon}}\langle m,x\rangle)dy_{i_{1}}\wedge...dy_{i_{n}}$, where $m=m(\gamma)$ defines the sector. It is easy to see that the differential $\overline{\partial}^{\prime}$ on Dolbeault forms on $X$ corresponds to the de Rham differential $d$ on $\Omega^{\ast}(Y)$. In this way we obtain an isomorphism of complexes $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}\simeq\underline{\Omega}^{\ast}_{Y,y}\otimes{{\bf C}}$.

The set of exponents appearing in the expansion of $\alpha_{m}$ at $y$ corresponds to the spectrum $Sp_{y}(\alpha)$ considered in the non-archimedean case.

Now we can define a semigroup $\varphi^{t}:Hom_{Dolb}(E_{1},E_{2})\to Hom_{Dolb}(E_{1},E_{2}),0\leq t<+\infty$. This is an analog of the semigroup $\phi^{t}$ in the non-archimedean case. First, we identify the sector $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}$ with $\underline{\Omega}^{\ast}(Y,Hom(\rho_{1},\rho_{2}))_{y}$ as above. Let us recall from the non-archimedean part, that to the homotopy class of a path $\gamma$ we canonically associated a closed $1$-form $\mu_{\gamma}=\int_{\gamma}\omega$, where $\omega$ is the symplectic form on $X^{\vee}$. Using the Riemannian metric $g_{Y}$ on $Y$ we assign to $\mu$ a vector field $\xi_{\gamma}$ on $Y$. In a local trivialization it is given by $grad((f_{1}-f_{2}+\langle m(\gamma),\cdot\rangle)/\varepsilon)$. Then the infinitesimal action of $\varphi^{t}$ is defined as the Lie derivative $Lie_{\xi_{\gamma}}$. Different Fourier components (sectors) move on $Y$ with with different speeds in different directions. Hence the picture is more complicated than in the case of Morse theory.

One can show that the generator $\Delta={d\over{dt}}|_{{t=0}}\varphi^{t}$ is a second order differential operator on $Hom_{Dolb}(E_{1},E_{2})$. When $g_{Y}$ is a flat metric and $f_{1}=f_{2}=0$ one can find the following explicit formula for $\Delta$:

It seems plausible that there is an extension of the semigroup $\varphi^{t}=e^{t\Delta}$, $t\geq 0$, from $Hom_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ to the whole space of morphisms $Hom_{{\cal A}(Y)}(E_{1},E_{2})$. Notice that $\Delta$ is not self-adjoint, and its real part is not elliptic. Nevertheless, we expect that the semigroup operator $\varphi^{t}$ converges as $t\to+\infty$ to a “projector” as in the case of Morse theory.

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Addresses:

M.K.: IHES, 35 route de Chartres, F-91440, France

maxim@ihes.fr

Y.S.: Department of Mathematics, KSU, Manhattan, KS 66506, USA

soibel@math.ksu.edu