Instantons, Quivers and Noncommutative Donaldson-Thomas Theory

TL;DR

This paper develops a framework for computing noncommutative Donaldson-Thomas invariants for abelian orbifold singularities using instanton moduli spaces, quiver representations, and topological gauge theories.

Contribution

It introduces a novel construction of noncommutative DT invariants via quiver moduli spaces linked to orbifold singularities and instanton counting.

Findings

Constructed moduli spaces for noncommutative instantons.

Connected invariants to quiver representation theory.

Provided explicit examples with higher rank and compact geometries.

Abstract

We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analysing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson-Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.