Instanton counting and Donaldson–Thomas theory on toric Calabi–Yau four-orbifolds

We study rank $r$ cohomological Donaldson–Thomas theory on a toric Calabi–Yau orbifold of $\mathbb{C}^4$ by a finite abelian subgroup $\Gamma$ of $\mathsf{SU}(4)$, from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on $\mathbb{C}^4 / \Gamma$ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over $r$-vectors of $\Gamma$-coloured solid partitions. When the $\Gamma$-action fixes an affine line in $\mathbb{C}^4$, we exhibit the dimensional reduction to rank $r$ Donaldson–Thomas theory on the toric Kähler three-orbifold $\mathbb{C}^3 / \Gamma$. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds $\mathbb{C}^2 / \mathbb{Z}_n \times \mathbb{C}^2$ and $\mathbb{C}^3 / (\mathbb{Z}_2 \times \mathbb{Z}_2) \times \mathbb{C}$, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.