A wall-crossing formula for Gromov-Witten invariants under variation of git quotient

TL;DR

This paper establishes a quantum wall-crossing formula for Gromov-Witten invariants under changes in GIT quotients, with implications for crepant transformations and their invariants.

Contribution

It provides a quantum version of Kalkman's wall-crossing formula relating Gromov-Witten invariants across GIT quotients with different polarizations.

Findings

Proves a quantum wall-crossing formula for Gromov-Witten invariants.

Shows equivalence of graph Gromov-Witten potentials under crepant type wall-crossings.

Connects wall-crossing phenomena with the crepant transformation conjecture.

Abstract

We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants with smaller structure group. As an application, we show that the graph Gromov-Witten potentials of quotients related by wall-crossings of crepant type are equivalent up to a distribution in the quantum parameter that is almost everywhere zero. This is a version of the crepant transformation conjecture of Li-Ruan, Bryan-Graber, Coates-Ruan etc. in cases where the crepant transformation is obtained by variation of git.