Localized mirror functor constructed from a Lagrangian torus

TL;DR

This paper constructs a functor from the Fukaya category of a symplectic manifold to matrix factorizations using a Lagrangian torus, providing a new bridge between symplectic geometry and algebraic categories, with explicit applications to toric Fano manifolds.

Contribution

It introduces a canonical A-infinity functor from the Fukaya category to matrix factorizations based on the Floer potential, unifying Lagrangian Floer theory with mirror symmetry constructions.

Findings

Constructed a holomorphic Floer potential W for Lagrangian tori.

Developed a functor from Fukaya category to matrix factorizations.

Explicitly derived the mirror matrix factorization for the real locus of complex projective space.

Abstract

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A-infinity functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.