Family Floer cohomology and mirror symmetry

TL;DR

This paper explores how Strominger-Yau-Zaslow's geometric approach to mirror symmetry can be used to construct the mirror of a symplectic manifold with a Lagrangian torus fibration, focusing on the case without singular fibres.

Contribution

It provides a framework for constructing the mirror as a moduli space of objects in the Fukaya category, extending the geometric approach to mirror symmetry.

Findings

Constructs the mirror as a moduli space of Fukaya category objects

Provides a method to associate twisted coherent sheaves to Lagrangian submanifolds

Explains the mirror construction in the absence of singular fibres

Abstract

Ideas of Fukaya and Kontsevich-Soibelman suggest that one can use Strominger-Yau-Zaslow's geometric approach to mirror symmetry as a torus duality to construct the mirror of a symplectic manifold equipped with a Lagrangian torus fibration as a moduli space of simple objects of the Fukaya category supported on the fibres. In the absence of singular fibres, the construction of the mirror is explained in this framework, and, given a Lagrangian submanifold, a (twisted) coherent sheaf on the mirror is constructed.