Quantum operations on the ring of symmetric functions

TL;DR

This paper develops a new framework for defining and studying $K$-theoretic Gromov-Witten invariants for stable maps into classifying stacks and quotient stacks, using stability stratifications to handle non-finite evaluation morphisms.

Contribution

It introduces a novel approach to $K$-theoretic invariants via stability stratifications for moduli stacks of stable maps into stacks like $B\mathrm{GL}_N$ and $Z/\mathrm{GL}_N$, extending the theory to new settings.

Findings

Constructed stability stratifications for moduli stacks of stable maps.

Defined $K$-theoretic Gromov-Witten invariants in this new setting.

Provided new proper moduli spaces of gauged maps.

Abstract

We define a version of stable maps into the classifying stack $B\mathrm{GL}_N$, and develop a corresponding notion of $K$-theoretic Gromov-Witten invariants. In this setting, the evaluation morphisms are not of finite type; the definition of the $K$-theoretic invariants proceeds by constructing a stability stratification of the moduli stack. In the absence of markings, the semistable locus of the stratification recovers moduli spaces of bundles on nodal curves considered by Gieseker, Nagaraj-Seshadri, Schmitt and Kausz. We also define versions of stable maps into quotient stacks of the form $Z/\mathrm{GL}_N$, where $Z$ is a projective $\mathrm{GL}_N$-scheme. We construct corresponding stability stratifications, whose semistable loci provide new proper moduli spaces of gauged maps from a varying nodal curve into $Z/\mathrm{GL}_N$.