Quantum Kirwan for quantum K-theory

TL;DR

This paper develops a formal map in quantum K-theory relating equivariant quantum K-theory to the quantum K-theory of GIT quotients, with applications to toric stacks, wall-crossing, and invariance under crepant transformations.

Contribution

It introduces a new formal map in quantum K-theory for GIT quotients and proves invariance and wall-crossing formulas, extending the understanding of quantum K-theoretic invariants.

Findings

Constructed a formal map from equivariant quantum K-theory to quantum K-theory of GIT quotients.

Provided a presentation of quantum K-theory for smooth proper toric Deligne-Mumford stacks.

Proved wall-crossing formulas and invariance of certain K-theoretic Gromov-Witten invariants.

Abstract

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git quotient $QK(X//G)$ assuming the quotient $X//G$ is a smooth Deligne-Mumford stack with projective coarse moduli space. As an example, we give a presentation of the (possibly bulk-shifted) quantum K-theory of any smooth proper toric Deligne-Mumford stack with projective coarse moduli space. We also provide awall-crossing formula for the K-theoretic gauged potential under variation of git quotient, a proof of the invariance of certain K-theoretic Gromov-Witten invariants under (strong) crepant transformation assumptions, and a proof of a version of the abelian non-abelian correspondence.