Homological mirror symmetry and torus fibrations

TL;DR

This paper explores two key conjectures in Mirror Symmetry, proposing a geometric perspective on torus fibrations and attempting to prove the homological mirror conjecture, with results on Massey products for abelian varieties.

Contribution

It offers a new geometric approach to understanding torus fibrations and makes progress towards proving the homological mirror conjecture using these fibrations.

Findings

Description of mirror manifolds via differential geometry

Identification of Massey products on both sides of mirror symmetry

Progress towards proving the homological mirror conjecture in specific cases

Abstract

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than the one in terms of D-branes. Also we make an attempt to prove the homological mirror conjecture using the torus fibrations. In the case of abelian varieties, and for a large class of Lagrangian submanifolds, we obtain an identification of Massey products on the symplectic and holomorphic sides. Tools used in the proof are of a mixed origin: not so classical Morse theory, homological perturbation theory and non-archimedean analysis.