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 $\mathsf{\Gamma }$ of $\mathsf{S}\mathsf{U}(4)$, 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 ${\mathbb{C}}^4\mathrm{/}\mathsf{\Gamma }$ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over $r$-vectors of $\mathsf{\Gamma }$-coloured solid partitions. When the $\mathsf{\Gamma }$-action fixes an affine line in ${\mathbb{C}}^4$, we exhibit the dimensional reduction to rank $r$ Donaldson-Thomas theory on the toric Kahler three-orbifold ${\mathbb{C}}^3\mathrm{/}\mathsf{\Gamma }$. 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 ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n\times {\mathbb{C}}^2$ and ${\mathbb{C}}^3\mathrm{/}({\mathbb{Z}}_2\times {\mathbb{Z}}_2)\times \mathbb{C}$, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.