Quantum Kirwan morphism and Gromov-Witten invariants of quotients I

TL;DR

This paper introduces a quantum version of the Kirwan map linking equivariant quantum cohomology of a variety with the orbifold quantum cohomology of its GIT quotient, and demonstrates its compatibility with genus zero Gromov-Witten potentials.

Contribution

It constructs the quantum Kirwan map for smooth projective varieties with reductive group actions and proves its intertwining property with Gromov-Witten potentials.

Findings

Constructed the quantum Kirwan map for GIT quotients.

Proved the map intertwines genus zero Gromov-Witten potentials.

Established foundational results for quantum cohomology of quotients.

Abstract

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.