ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds

TL;DR

This paper develops an ADHM-like framework to classify invariant solutions in supersymmetric gauge theories on non-commutative four-dimensional space, connecting solid partitions to Donaldson-Thomas invariants on orbifolds.

Contribution

It introduces a novel ADHM-like quiver construction for 8d gauge theories and classifies solutions using coloured solid partitions, linking combinatorics with geometric invariants.

Findings

Explicit computation of partition functions on ${f C}^4$ and ${f C}^2\times({f C}^2/{\mathbb Z}_2)$

Conjectural formulas for orbifold Donaldson-Thomas invariants

Classification of invariant solutions via coloured solid partitions

Abstract

We study the moduli space of $SU(4)$ invariant BPS conditions in supersymmetric gauge theory on non-commutative ${\mathbb C}^4$ by means of an ADHM-like quiver construction and we classify the invariant solutions under the natural toric action in terms of solid partitions. In the orbifold case ${\mathbb C}^4/G$, $G$ being a finite subgroup of $SU(4)$, the classification is given in terms of coloured solid partitions. The statistical weight for their counting is defined through the associated equivariant cohomological gauge theory. We explicitly compute its partition function on ${\mathbb C}^4$ and ${\mathbb C}^2\times\left({\mathbb C}^2/{\mathbb Z}_2\right)$ which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.