We study rank r cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of 4 by a finite abelian subgroup Γ of 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 4 /Γ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over r-vectors of Γ-coloured solid partitions. When the Γ-action fixes an affine line in 4 , we exhibit the dimensional reduction to rank r Donaldson-Thomas theory on the toric Kähler three-orbifold 3 /Γ. 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 2 / n × 2 and 3 /( 2 × 2 ) × , finding perfect agreement with new mathematical results of Cao, Kool and Monavari.

This is a mini-review about generalized instantons of noncommutative gauge theories in dimensions 4, 6 and 8, with emphasis on their realizations in type II string theory, their geometric interpretations, and their applications to the enumerative geometry of non-compact toric varieties.

This is a mini-review about generalized instantons of noncommutative gauge theories in dimensions 4, 6 and 8, with emphasis on their realizations in type II string theory, their geometric interpretations, and their applications to the enumerative geometry of non-compact toric varieties.