Here we will define dg-category ${\cal C}_{unram}(Y)$ and the fully faithful embedding $F$ of this category to ${\cal C}(Y)$.

In the Appendix we will explain the conventional picture for the mirror symmetry functor in case of complex numbers. There we will use a kind of Fourier-Mukai transform along fibers of the torus fibration. The kernel of this transform is an analog of Poincaré bundle. If one starts with a local system on a Lagrangian section of $p^{\vee}:X^{\vee}\to Y$ then the transform makes from it a smooth bundle on $X$ with the connection which is flat in the anti-holomorphic directions. In other words, one gets a holomorphic bundle on $X$.

These considerations cannot be literally repeated in the non-archimedean case, because “holomorphic” considerations do not work. One can obtain the same result in the following way. Let $(L,\rho)$ be an object of the category $FO(X^{\vee})$ such that $rank(\rho)=1$ and the projection $L\to Y$ is one-to-one map. The manifold $L$ is locally given by the graph of $df\,(mod\,(T_{Y}^{{\bf Z}})^{\vee})$, where $f$ is a smooth function on $Y$. To such an object we assign a sheaf of rank one ${\cal O}_{Y}$-modules $F(L,\rho)$. For sufficiently small open $U\subset Y$ and chosen $f\in C^{\infty}(U)$ the sheaf $F(L,\rho)_{|U}$ is identified with ${{\cal O}_{Y}}_{|U}$. Change $f\mapsto f+l$, where $l\in Af_{Y}(U)$ leads to the change of the trivialization of $F(L,\rho)_{|U}$ as $1_{U}\mapsto exp(l)\,1_{U}$ (here $1_{U}\in{\cal O}_{Y}(U)$ is the identity function). If $rank(\rho)$ is greater than one, we decompose $\rho_{|U}$ for small $U\subset Y$ into the sum of rank one local systems and then apply the construction. Analogously, if the covering $L\to Y$ has more than one leaf, we apply the previous construction to each leaf of the covering and then take the direct sum.

We will loosely call $F$ the mirror symmetry functor on objects. The category ${\cal C}_{unram}(Y)$ is defined as the dg-category whose class of objects is $Ob(FO(X^{\vee}))$, and the spaces of morphisms are

$$Hom_{{\cal C}_{unram}(Y)}((L_{1},\rho_{1}),(L_{2},\rho_{2})):=Hom_{{\cal C}(Y)}(F((L_{1},\rho_{1}),F(L_{2},\rho_{2}))$$

The functor $F$ on morphisms is defined in the obvious way, as the identity map.

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.

The whole approach here is parallel to the one from Section 6, so we will omit the details. In the previous subsection we defined the semigroup $\psi^{t}$ acting on the sections with compact support $\Gamma_{c}(P(L_{1},L_{2}),\underline{Hom}^{alg}(E_{1},E_{2}))$. This action corresponds to the action of the semigroup $\phi^{t}$ on the space of morphisms $Hom_{{\cal C}(Y)}(E_{1},E_{2})$. Similarly to the case of Morse theory (Section 6) one proves the following result.

The limit is not difficult to describe in terms of the gradient flow generating $\psi^{t}$. Using the fact that $\psi^{t}$ moves the spectrum of a morphism to $-\infty$, one can prove similarly to the Section 6 that the limit $\psi^{\infty}(\beta)$ belongs to a finite-dimensional ${{\bf C}}_{\varepsilon}$-vector space generated by the distributions corresponding the unstable manifolds $U_{x}\subset P(L_{1},L_{2}),x\in L_{1}\cap L_{2}$. Clearly, the map $\beta\mapsto\psi^{\infty}(\beta)$ extends to the completion with respect to the filtration. It descends to the map $\alpha\mapsto\phi^{\infty}(\alpha)$, where $\alpha\in Hom_{{\cal C}(Y)}(E_{1},E_{2})$. The image of $\phi^{\infty}$ belongs to the space isomorphic to $Hom_{FO(X^{\vee})}((L_{1},\rho_{1}),(L_{2},\rho_{2}))$.

We can repeat the arguments from the Morse theory (see Section 6). We define the $A_{\infty}$-pre-category ${\cal C}_{unram,0}(Y)$ similarly to the category $DR_{0}(Y)$ from Section 6. It is $A_{\infty}$-equivalent to ${\cal C}_{unram}(Y)$. By definition the spaces of morphisms of ${\cal C}_{unram,0}(Y)$ are dg-modules over the dg-algebra ${{\bf C}}_{\varepsilon}\widehat{\otimes}\Omega_{0}^{\ast}$, where $\Omega_{0}^{\ast}$ is the dg-algebra of germs of differential forms at $0\in{{\bf R}}_{\geq 0}$. Compositions of morphisms in ${\cal C}_{0}(Y)$ are linear with respect to the dg-module structure. Imposing transversality conditions on ${\cal C}_{unram,0}(Y)$ to be the same as in $FO(X^{\vee})$, we obtain an $A_{\infty}$-equivalent $A_{\infty}$-pre-category ${\cal C}_{unram,0}^{tr}(Y)$.

Using homological perturbation theory (projectors and homotopies are defined by means of the semigroup) similarly to Section 6, we construct an analog of the category $DR_{0}^{tr,\Pi}(Y)$. It is an $A_{\infty}$-pre-category denoted by ${\cal C}_{unrum,0}^{tr,\Pi}(Y)$, with the spaces of morphisms which are completed tensor products of $\Omega_{0}^{\ast}$ with finite-dimensional ${{\bf C}}_{\varepsilon}$-vector spaces, spanned by the “smoothenings” of the unstable currents $[U_{x}]$ (cf. Section 6). By definition, it has the same transversality conditions as the category $FO(X^{\vee})$, and the spaces of morphisms are naturally quasi-isomorphic to the corresponding spaces of morphisms in $FO(X^{\vee})$ (compare with the Section 6.6). Similarly to the Section 6 we see that the $A_{\infty}$-structure on ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ is equivalent to the one on $FO(X^{\vee})$. More precisely, we have a natural map from the space $Hom_{FO(X^{\vee})}((L_{1},\rho_{1}),(L_{2},\rho_{2}))$ (it is defined in terms of the Morse theory) to the space $Hom_{{\cal C}_{unram,0}^{tr,\Pi}(Y)}(F(L_{1},\rho_{1}),F(L_{2},\rho_{2}))$ (it is defined in terms of de Rham differential forms on $Y$). Thus we have defined the mirror symmetry functor $F$ on morphisms. Let us call the corresponding map $\nu_{X_{1},X_{2}}$ for $X_{i}=(L_{i},\rho_{i}),i=1,2$. The proof of the following proposition is similar to its analog from Section 6.6.

Thus, $A_{\infty}$-pre-categories ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ and $FO(X^{\vee})$ are equivalent. By the same arguments as in the Morse theory section we see that ${\cal C}_{unram}(Y)$ and ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ are also equivalent. Finally, applying functor $F$, we get our main result.

This is the version of homological mirror symmetry we promised to prove.