Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories

TL;DR

This paper develops a comprehensive framework for understanding how equivariant K-theoretic enumerative invariants of moduli stacks change across walls, extending Joyce's homological wall-crossing to K-theory with equivariance, using operatinal K-homology as a vertex algebra.

Contribution

It introduces a novel approach by lifting Joyce's wall-crossing to K-theory and incorporating equivariance through the operational K-homology's structure as a vertex algebra.

Findings

Framework for wall-crossing in equivariant K-theory

Operational K-homology forms an equivariant multiplicative vertex algebra

Extension of Joyce's homological wall-crossing to K-theoretic setting

Abstract

We provide a general framework for wall-crossing of equivariant K-theoretic enumerative invariants of appropriate moduli stacks $\mathfrak{M}$, by lifting Joyce's homological universal wall-crossing arXiv:2111.04694 to K-theory and to include equivariance. The primary new tool is that the operational K-homology of $\mathfrak{M}$ is an equivariant multiplicative vertex algebra.