Quantum operations on the ring of symmetric functions

TL;DR

The paper builds a comprehensive moduli-theoretic framework for stable maps into classifying stacks $B\mathrm{GL}_N$ and quotient stacks $Z/\mathrm{GL}_N$, introducing a Theta-stratification approach that yields proper moduli spaces and well-defined $K$-theoretic Gromov–Witten invariants. It introduces pure/Gieseker bundles with marked boundary data, proves a Theta-stratification for the corresponding moduli, and defines a level line bundle $\mathcal{L}_{\mathrm{lev}}$ that governs gluing and pairings. A Fourier–Mukai-type transform $\Omega^a_{g,n,p}(N,d)$ is constructed, shown to preserve perfect complexes (admissibility), and extended to projective targets via stacks of Gieseker maps, with explicit analysis in the $N=1$, $g=0$ case. The framework is then adapted to gauged maps into projective targets $Z$, establishing projectivity and stratifications for the moduli spaces, providing a route toward a broader gauged Gromov–Witten theory with potential connections to wall-crossing and quantum invariants. Overall, the work develops robust moduli-theoretic tools (Theta-stratifications, level line bundles, and Kontsevich-type compactifications) to define and control $K$-theoretic invariants for gauged maps, with systematic extensions to general projective targets and low-rank explicit computations.

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 $Z/\mathrm{GL}_N$.

Figures (6)

• Figure 1: Figure 1
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Theorems & Definitions (153)

• Theorem A: = \ref{['T:admissibility']}
• Definition 1.1
• Definition 1.2: Pure Gieseker bundles
• Theorem B
• Definition 2.1: Stack of marked prestable curves
• Definition 2.2: Stack of polarized marked curves
• Proposition 2.3
• proof
• Definition 2.4: Stack of marked sheaves
• Definition 2.5: Stack of pure marked sheaves