Quantum cohomology and toric minimal model programs

TL;DR

This paper develops a quantum cohomology framework for compact toric orbifolds, extending classical results and linking algebraic structures with geometric properties like Hamiltonian non-displaceable tori.

Contribution

It constructs a canonical isomorphism between a formal Batyrev ring and quantum orbifold cohomology, generalizing previous results to orbifolds using a toric minimal model program.

Findings

Established a quantum version of the Danilov-Jurkiewicz presentation.

Proved a quantum Kirwan surjectivity and dimension equality.

Linked quantum cohomology decomposition to tmmp singularities and non-displaceable tori.

Abstract

We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental, Iritani, and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity and an equality of dimensions deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each giving rise to a collection of Hamiltonian non-displaceable tori.