Tetrahedron Instantons on Orbifolds

Abstract

Given a homomorphism $τ$ from a suitable finite group $\mathsfΓ$ to $\mathsf{SU}(4)$ with image $\mathsfΓ^τ$, we construct a cohomological gauge theory on a noncommutative resolution of the quotient singularity $\mathbb{C}^4/\mathsfΓ^τ$ whose BRST fixed points are $\mathsfΓ$-invariant tetrahedron instantons on a generally non-effective orbifold. The partition function computes the expectation values of complex codimension one defect operators in rank $r$ cohomological Donaldson-Thomas theory on a flat gerbe over the quotient stack $[\mathbb{C}^4/\,\mathsfΓ^τ]$. We describe the generalized ADHM parametrization of the tetrahedron instanton moduli space, and evaluate the orbifold partition functions through virtual torus localization. If $\mathsfΓ$ is an abelian group the partition function is expressed as a combinatorial series over arrays of $\mathsfΓ$-coloured plane partitions, while if $\mathsfΓ$ is non-abelian the partition function localizes onto a sum over torus-invariant connected components of the moduli space labelled by lower-dimensional partitions. When $\mathsfΓ=\mathbb{Z}_n$ is a finite abelian subgroup of $\mathsf{SL}(2,\mathbb{C})$, we exhibit the reduction of Donaldson-Thomas theory on the toric Calabi-Yau four-orbifold $\mathbb{C}^2/\,\mathsfΓ\times\mathbb{C}^2$ to the cohomological field theory of tetrahedron instantons, from which we express the partition function as a closed infinite product formula. We also use the crepant resolution correpondence to derive a closed formula for the partition function on any polyhedral singularity.