Quantum Kirwan morphism and Gromov-Witten invariants of quotients III

TL;DR

This paper develops a quantum version of the Kirwan map linking equivariant and orbifold quantum cohomology for GIT quotients, and provides formulas for solutions to quantum differential equations.

Contribution

It constructs a quantum Kirwan map for GIT quotients and proves its compatibility with Gromov-Witten potentials, extending previous work in the field.

Findings

Construction of a quantum Kirwan map for GIT quotients

Proof that the map intertwines genus zero Gromov-Witten potentials

Formula for solutions to quantum differential equations in terms of localized gauged potentials

Abstract

This is the third in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those of Givental, Lian-Liu-Yau, Coates-Corti-Iritani-Tseng and Ciocan-Fontanine-Kim.