Instanton counting and wall-crossing for orbifold quivers

TL;DR

This paper explores the enumeration of instanton solutions in noncommutative gauge theories related to orbifold singularities, extending to refined invariants and analyzing wall-crossing phenomena through quiver representations.

Contribution

It introduces a framework for computing higher-rank noncommutative Donaldson-Thomas invariants using virtual instanton quivers and connects wall-crossing behavior with the McKay correspondence.

Findings

Extended instanton counting to refined and motivic invariants.

Defined virtual instanton quivers for wall-crossing analysis.

Linked formalism with quantum monodromy and cluster algebras.

Abstract

Noncommutative Donaldson-Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative N=2 gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated to the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson-Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.