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

We study the moduli space of SU (4) invariant BPS conditions in supersymmetric gauge theory on non-commutative C4 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 C4/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 several terms of the partition function expansion in the instanton counting parameter on C4 and C2 x (C2/Z2), which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.