Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds

TL;DR

This paper develops a combinatorial and geometric framework for computing Donaldson-Thomas invariants on toric Calabi-Yau four-orbifolds using instanton counting, with explicit formulas and connections to known mathematical results.

Contribution

It introduces a new description of moduli spaces of noncommutative instantons on toric Calabi-Yau four-orbifolds and derives explicit partition functions with conjectured closed-form formulas.

Findings

Computed orbifold instanton partition functions as combinatorial series.

Established dimensional reduction to Donaldson-Thomas theory on three-orbifolds.

Conjectured infinite product formulas matching mathematical results.

Abstract

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 $\mathsf{SU}(4)$, 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 $\mathbb{C}^4/\mathsf{\Gamma}$ 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 $\mathbb{C}^3/\mathsf{\Gamma}$. 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 $\mathbb{C}^2/\mathbb{Z}_n\times\mathbb{C}^2$ 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.