Lagrangian Floer theory and mirror symmetry on compact toric manifolds

TL;DR

This paper establishes a deep connection between Lagrangian Floer theory and mirror symmetry on compact toric manifolds by constructing an isomorphism between their Frobenius manifold structures, linking quantum cohomology and singularity theory.

Contribution

It introduces a natural isomorphism between quantum cohomology and singularity theory for toric manifolds using Floer cohomology and open-closed Gromov-Witten theory.

Findings

Proves the isomorphism between Frobenius manifolds of quantum cohomology and singularities.

Uses open-closed Gromov-Witten theory of one-loop to establish the correspondence.

Links Floer cohomology deformed by ambient cycles to mirror symmetry structures.

Abstract

In this paper we study Lagrangian Floer theory on toric manifolds from the point of view of mirror symmetry. We construct a natural isomorphism between the Frobenius manifold structures of the (big) quantum cohomology of the toric manifold and of Saito's theory of singularities of the potential function constructed in \cite{fooo09} via the Floer cohomology deformed by ambient cycles. Our proof of the isomorphism involves the open-closed Gromov-Witten theory of one-loop.