Hodge-Theoretic Mirror Symmetry for Toric Stacks

TL;DR

This paper establishes a mirror symmetry correspondence for toric stacks, linking their quantum cohomology to Landau-Ginzburg models through Hodge-theoretic methods, and provides explicit algebraic and combinatorial descriptions.

Contribution

It proves the isomorphism between the quantum D-module of toric stacks and the Saito structure of the mirror, extending mirror symmetry to a Hodge-theoretic framework.

Findings

Isomorphism between quantum D-module and Saito structure established

Provides a GKZ-style presentation of quantum D-module

Demonstrates convergence of mirror and quantum cohomology in big and equivariant cases

Abstract

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.