Let $E_{i}=F(L_{i},\rho_{i}),i=1,2$ be locally free ${\cal O}_{Y}$-modules (i.e. vector bundles) corresponding to objects $(L_{i},\rho_{i})\in FO(X^{\vee}),i=1,2$. For any $\alpha\in Hom_{{\cal C}(Y)}(E_{1},E_{2})$ and a point $y\in Y$ we will define the spectrum of $\alpha$ at $y$ as a certain (at most countable) discrete set of real numbers with finite multiplicities.

Let us assume first that $\rho_{i},i=1,2$ are trivial rank one local systems on $L_{i},i=1,2$, and $L_{i},i=1,2$ are unramified coverings of $Y$. For a sufficiently small open set $U$ containing $y$ we can write in local coordinates $L_{i}=graph(df_{i})\,(mod\,(T_{Y}^{{\bf Z}})^{\vee}),i=1,2$ for smooth functions $f_{i}:Y\to{{\bf R}},i=1,2$. Restriction to a small open set $U$ of a morphism $\alpha\in Hom_{{\cal C}(Y)}(E_{1},E_{2})(U)=\widehat{\Omega}_{Y}^{\ast}(U)$ can be identified with the infinite series $\alpha=\sum_{i_{1},...,i_{n}}c_{i_{1}...i_{n}}z_{1}^{i_{1}}...z_{n}^{i_{n}}$, where $c_{i_{1}...i_{n}}=\sum_{j}c_{j,i_{1}...i_{n}}e^{-\lambda_{j,i_{1}...i_{n}}/\varepsilon}$ and $c_{j,i_{1}...i_{n}}\in{\Omega}_{Y}^{\ast}(U)$.

We define the spectrum of $\alpha$ at $y\in U$ as the set of real numbers (with multiplicities)

$$Sp_{y}(\alpha)=\{-\lambda_{j,i_{1}...i_{n}}+\sum_{1\leq k\leq n}i_{k}y_{k}+f_{1}(y)-f_{2}(y)\}\,,$$

where the germ of $c_{j,i_{1}...i_{n}}$ at $y$ is not equal to zero. One can check that $Sp_{y}(\alpha)$ is well-defined (i.e. does not depend on the local trivialization), and has the only limiting point at $s=-\infty$.

In the general case of higher rank local systems and Lagrangian manifolds which are unramified coverings of $Y$, we decompose $E_{i},i=1,2$ locally near $y\in Y$ into the direct sum of trivial rank one ${\cal O}_{Y}$-modules. The spectrum of a morphism at the point $y$ is then defined as the union of the spectra of morphisms between corresponding line bundles.

Let us consider a subspace $Hom_{{\cal C}(Y)}^{alg}(E_{1},E_{2})\subset Hom_{{\cal C}(Y)}(E_{1},E_{2})$ of algebraic morphisms. It consists of finite sums (both in $z_{i}$ and $e^{-\lambda_{j,i_{1}...i_{n}}/\varepsilon}$). It is dense in the space of all morphisms (analytic functions can be approximated by Laurent polynomials). Moreover, the space $Hom_{{\cal C}(Y)}(E_{1},E_{2})$ coincides with the completion of $Hom_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$ with respect to the ${{\bf R}}$-filtration introduced above.

There is a $1$-parameter semigroup $\phi^{t},t\geq 0$ acting on $Hom_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$. In local coordinates $\phi^{t}$ acts on the coefficients $c_{j,i_{1}...i_{n}}$ by moving them along the gradient flow of $f_{1}-f_{2}$. In order to define it globally we need to describe the space $Hom_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$ in geometric terms. It will be done below.

Given two Lagrangian submanifolds $L_{i}\subset X^{\vee},i=1,2$ as above, a point $y\in Y$, two points $x_{i}\in L_{i},i=1,2$ such that $p^{\vee}(x_{i})=y$, we define a set $P(L_{1},L_{2},y)$ of homotopy classes of paths $\gamma\in(p^{\vee})^{-1}(y)$ starting at $x_{1}$ and ending at $x_{2}$. Each homotopy class contains a unique geodesic in the flat metric on the torus. We define the space $P(L_{1},L_{2})=\sqcup_{y\in Y}P(L_{1},L_{2},y)$. It carries an obvious topology such that the natural projection $\pi:P(L_{1},L_{2})\to Y$ is an unramified covering with countable fibers. Using the symplectic form $\omega$ on $X^{\vee}$ we define a closed $1$-form $\mu$ on $P(L_{1},L_{2})$ by the formula $\mu=\int_{\gamma}\omega$. Locally on $Y$ we have: $L_{i}=df_{i}(mod\,(T_{Y}^{{\bf Z}})^{\vee}),i=1,2$ where $f_{i}:Y\to{{\bf R}}$ are smooth functions. Then locally on $P(L_{1},L_{2})$ we have: $\mu=d(f_{1}-f_{2}+l)$, where $l$ is a local section of the pullback of the sheaf $Aff_{Y}$. Clearly the function $l$ is defined up the adding of a real constant. Thus obtain an ${{\bf R}}$-torsor on $P(L_{1},L_{2})$. Using the embedding ${{\bf R}}\to{{\bf C}}_{\varepsilon}^{\ast}$, $\lambda\mapsto exp(\lambda/\varepsilon)$ we get a ${{\bf C}}_{\varepsilon}^{\ast}$-torsor, which defines a local system ${{\bf C}}_{\varepsilon}^{tw}$ of $1$-dimensional ${{\bf C}}_{\varepsilon}$-modules over $P(L_{1},L_{2})$. Fibers of ${{\bf C}}_{\varepsilon}^{tw}$ carry natural filtrations. Indeed, in a neighborhood of a point $(x_{1},x_{2},\gamma,y)\in P(L_{1},L_{2})$ we can choose a smooth function $f=f_{1}-f_{2}+l$ such that $\mu=df$. It defines a local trivialization of ${{\bf C}}_{\varepsilon}^{tw}$. In this trivialization the filtration is defined for $h\in{{\bf C}}_{\varepsilon}$ by the condition $v(h)(y)+f(y)\leq s,s\in{{\bf R}}$, where $v$ is the valuation. We define a subsheaf ${{\bf C}}_{\varepsilon}^{tw,alg}$ of ${{\bf C}}_{\varepsilon}^{tw}$ by the requirement that in a local trivialization it is a subsheaf of finite sums of exponents.

Notice that there are natural projections $pr_{i}:P(L_{1},L_{2})\to L_{i},i=1,2$. Having local systems $\rho_{i}$ on $L_{i},i=1,2$ we define local systems $\widehat{\rho}_{i},i=1,2$ on $P(L_{1},L_{2})$ as pullbacks with respect to $pr_{i},i=1,2$.

On $P(L_{1},L_{2})$ we define a sheaf $\underline{Hom}^{alg}(E_{1},E_{2})$ ($E_{i},i=1,2$ were defined previously) such as follows: $\underline{Hom}^{alg}(E_{1},E_{2})={{\bf C}}_{\varepsilon}^{tw,alg}\otimes(\widehat{\rho}_{1})^{\ast}\otimes\widehat{\rho}_{2}\otimes\underline{\Omega}_{P(L_{1},L_{2})}^{\ast}$, where $\underline{\Omega}_{P(L_{1},L_{2})}^{\ast}$ is the sheaf of differential forms. We endow stalks of $\underline{Hom}^{alg}(E_{1},E_{2})$ with ${{\bf R}}$-filtrations induced by the filtration on ${{\bf C}}_{\varepsilon}^{tw}$ and trivial filtrations on the other tensor factors.

Let $\pi_{!}$ denotes the functor of direct image with compact support. Then $\pi_{!}(\underline{Hom}^{alg}(E_{1},E_{2}))=\pi_{!}({{\bf C}}_{\varepsilon}^{tw}\otimes\widehat{\rho}_{1}^{\ast}\otimes\widehat{\rho}_{2})\otimes\underline{\Omega}_{Y}^{\ast}$, where the last tensor factor is the sheaf of de Rham differential forms on $Y$.

We can identify ${{\bf Z}}^{n}$ with $H_{1}(T^{n},{{\bf Z}})$, and the latter group naturally acts on homotopy classes of paths $\gamma$. On the other hand, the group ring of ${{\bf Z}}^{n}$ over ${{\bf C}}_{\varepsilon}$ can be identified with the ring of Laurent polynomials ${{\bf C}}_{\varepsilon}[z_{1}^{\pm 1},...,z_{n}^{\pm 1}]$. Let ${{\bf C}}_{\varepsilon}^{alg}\subset{{\bf C}}_{\varepsilon}$ be the subring of finite sums of exponents. It is easy to see that the structure of ${{\bf C}}_{\varepsilon}^{alg}[{{\bf Z}}^{n}]$-module on the sections of $\underline{Hom}^{alg}(E_{1},E_{2})$ corresponds to the structure of ${{\bf C}}_{\varepsilon}^{alg}[z_{1}^{\pm 1},...,z_{n}^{\pm 1}]$-module on its image under $\pi_{!}$. Using this observation one can prove that

$$Hom_{{\cal C}(Y)}^{alg}(E_{1},E_{2})\simeq\Gamma(Y,\pi_{!}(\underline{Hom}^{alg}(E_{1},E_{2})))=\Gamma_{c}(P(L_{1},L_{2}),\underline{Hom}^{alg}(E_{1},E_{2})),$$

where the isomorphism is induced by the natural morphism of sheaves

$$\pi_{!}(\underline{Hom}^{alg}(E_{1},E_{2}))\to\underline{Hom}_{{\cal C}(Y)}^{alg}(E_{1},E_{2})\,.$$

Here $\Gamma_{c}$ refers to the functor of sections with compact support.

Using the metric on $Y$ we assign to the $1$-form $\mu$ a vector field $\xi$ on $P(L_{1},L_{2})$. Locally $\xi$ is the generator of the gradient flow of $f_{1}-f_{2}+l$. It is not difficult to show that there is no trajectory of the flow which goes to infinity for a finite time. Therefore the vector field $\xi$ generates a $1$-parameter semigroup $\psi^{t}$ acting on $P(L_{1},L_{2})$. The following result is easy to prove.

Functor $\pi_{!}$ is compatible with the filtrations on the stalks of sheaves $\underline{Hom}_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$ and $\underline{Hom}^{alg}(E_{1},E_{2})$. It is easy to see that the completion of stalks of the former with respect to the filtration induced from the one on $\underline{Hom}^{alg}(E_{1},E_{2})$ coincides with $Hom_{{\cal C}(Y)}(E_{1},E_{2})$. Since the semigroup $\psi^{t}$ decreases the filtration, the semigroup $\phi^{t}$ extends continuously to the completion with respect to the filtration. Thus the following proposition holds.