Mathematically mirror symmetry can be interpreted in many ways. In this paper we will make a bridge between two approaches, the homological mirror symmetry ([Ko]) and the duality between torus fibrations (a version of Strominger-Yau-Zaslow conjecture, see [SYZ]).

The mirror symmetry is a duality between Calabi-Yau manifolds, i.e. complex manifolds with Kähler metrics with vanishing Ricci curvarure. In fact, it is rather duality not between individual manifolds, but between manifolds in certain “degenerating” families (“large complex structure limit” and “large symplectic structure limit”). In this paper we propose some conjectures of differential-geometric nature about the degenerations. In particular, we assume that in the limit both dual manifolds $X$ and $X^{\vee}$ become fiber bundles with toroidal fibers over the same base $Y$ (see Section 3). The manifold $Y$ is a (real) Riemannian manifold whose dimension is half of the dimension of $X$ and $X^{\vee}$. Also, manifold $Y$ carries a rich geometric structure which contains as a part a “combinatorial” data, so-called integral affine structure. This picture is partially motivated by the classical theory of collapsing of general Riemannian manifolds (see [CG]). Another origin of our geometric conjectures is the [SYZ] version of mirror duality. We recast it in somewhat different terms in Section 2, devoted to the moduli space of conformal field theories and its natural compactification. In a recent preprint [GW] similar differential-geometric conjectures were proposed and verified in the case of degenerating K3-surfaces.

The homological mirror symmetry conjecture (proposed in [Ko]) is a statement about an equivalence between two $A_{\infty}$-categories: the (derived) category of coherent sheaves on a Calabi-Yau manifold $X$ and the Fukaya category of the dual Calabi-Yau manifold $X^{\vee}$. The former is defined in holomorphic (or algebraic) terms, the latter is defined in terms of symplectic geometry.

We apply the geometric picture of limits of Calabi-Yau metrics to the homological mirror conjecture. The Fukaya category $F(X^{\vee},\omega^{\vee})$ of a symplectic manifold $(X^{\vee},\omega^{\vee})$, with $[\omega^{\vee}]\in H^{2}(X^{\vee},{{\bf Z}})$, and its degeneration are defined as $A_{\infty}$-categories over the field of Laurent formal power series ${{\bf C}}((q))$. The parameter $q$ enters in the story when one writes higher compositions, which have expressions $q^{\int_{\beta}\omega^{\vee}}$ as coefficients, $\beta\in H_{2}(X^{\vee},{{\bf Z}})$. We can set $q=exp(-1/\varepsilon),\varepsilon\to 0$, where the parameter $\varepsilon$ corresponds to the rescaling of the symplectic form: $\omega^{\vee}\mapsto\omega^{\vee}/\varepsilon$. If $[\omega^{\vee}]$ does not belong to $H^{2}(X,{{\bf Z}})$, one can work over the field ${{\bf C}}_{\varepsilon}:=\{\sum_{i\geq 0}a_{i}e^{-\lambda_{i}/\varepsilon}|\,a_{i}\in{{\bf C}},\lambda_{i}\in{{\bf R}},\lambda_{i}\to+\infty\}$.

In the picture of torus fibration, a full subcategory of the limiting Fukaya category can be described in terms of the Morse theory on the base of the torus fibration. The higher products giving the $A_{\infty}$-structure can be written as sums over sets of planar trees. In the case of cotangent bundle (instead of the torus filtration) this description was proposed earlier by Fukaya and Oh (see [FO]).

On the holomorphic side of mirror symmetry, the degeneration of the dual family $X_{q}$ is described in non-archimedean terms: we have a Calabi-Yau manifold ${\cal X}_{mer}$ over the field ${{\bf C}}_{q}^{mer}$ of germs at $q=0$ of meromorphic functions. Changing scalars, we get a Calabi-Yau manifold ${\cal X}_{form}$ over the local field of Laurent series ${{\bf C}}((q))$. Let us call this degeneration picture analytic. There is a description of some algebraic Calabi-Yau manifolds over arbitrary local fields (complete with respect to discrete valuations) in terms of real $C^{\infty}$-manifolds with integral affine structures. We expect that differential-geometric and (non-archimedean) analytic pictures of the degeneration are equivalent. In this paper we discuss the relationship between integral affine manifolds and varieties over non-archimedean fields only in the simplest case of flat tori and abelian varieties. The general case will be described elsewhere.

The homological mirror conjecture says that the Fukaya category $F(X^{\vee},\omega^{\vee})$ is equivalent (as an $A_{\infty}$-category over ${{\bf C}}((q))\,$) to the derived category of coherent sheaves $D^{b}({\cal X}_{form})$. Apparently, it implies well-known numerical predictions for the numbers of rational curves on a Calabi-Yau manifold (genus zero Gromov-Witten invariants).

Using the assumptions about the collapse, we offer in the paper a general approach to the proof of Homological Mirror Conjecture and apply it in the case when the torus fibration has no singularities. This happens in the case of abelian varieties. Also, we deal not with all objects in the Fukaya category, but with a certain subclass. In general, one should investigate the input of singularities of the base of torus fibration.

It should be clear from the above discussion that the non-archimedean analysis plays an important role in the formulation and proof of the main result. Analytic picture of the degeneration seems to be related to the theory of rigid analytic spaces in the version of Berkovich (see [Be]). In particular, there is a striking similarity between the base of torus fibration and a certain canonically defined subset (see 3.3) of the skeleton of an analytic space introduced in [Be]. This subject definitely deserves further investigation.

In Section 2 we discuss motivations from the Conformal Field Theory. In Section 3 we formulate the conjectures about analytic and geometric pictures of the large complex structure limit. In Section 4 we describe a general framework of $A_{\infty}$-pre-categories adapted to the transversality problem in the definition of the Fukaya category. Section 5 is devoted to the Fukaya category and its degeneration. The reader will notice an advantage of working over the field of Laurent power series: one can consider all local systems over Lagrangian submanifolds, while in the conventional approach unitarity of the holonomy is required. Section 6 is devoted to the $A_{\infty}$-category of smooth functions introduced by Fukaya (and then studied by Fukaya and Oh in [FuO]). We prove that this $A_{\infty}$-category has very simple de Rham model. This part of the paper can be read independently of the rest. On the other hand, the technique of the proof will be used later in the paper. One important technical tool is an explicit $A_{\infty}$-structure on a subcomplex of a differential-graded algebra (see [GS], [Me]). We restate the formulas from [Me] in term of sums over a set of planar trees. The proof of the equivalence of Morse and de Rham $A_{\infty}$-categories uses the technique of [HL]. Section 7 is devoted to the analytic side of the homological mirror conjecture. We give a construction of mirror symmetry functor for torus fibrations in terms of the non-archimedean geometry. The use of non-archimedean analysis allows us to avoid problems with convergence of series in the definition of the Fukaya category. In Section 8 we construct an $A_{\infty}$-pre-category which is equivalent to a full $A_{\infty}$-subcategory of the derived category coherent sheaves on the Calabi-Yau manifold over ${{\bf C}}_{\varepsilon}$. Similarly to the comparison of Morse and de Rham pictures, we will prove that this category is also equivalent to an $A_{\infty}$-subcategory of the Fukaya category of the mirror dual torus fibration. In Appendix (Section 9) we describe the analogs of our constructions in the case of complex geometry.