Quantum operations on the ring of symmetric functions

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 $Z/{\mathrm{GL}}_N$.

63 pages with expository endnotes. Comments are welcome