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.