Tetrahedron Instantons on Orbifolds

Given a homomorphism $\tau$ from a suitable finite group $\mathsf{\Gamma}$ to $\mathsf{SU}(4)$ with image $\mathsf{\Gamma}^\tau$, we construct a cohomological gauge theory on a noncommutative resolution of the quotient singularity $\mathbb{C}^4/\mathsf{\Gamma}^\tau$ whose BRST fixed points are $\mathsf{\Gamma}$-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 $[\mathbb{C}^4/\,\mathsf{\Gamma}^\tau]$. We describe the generalized ADHM parametrization of the tetrahedron instanton moduli space, and evaluate the orbifold partition functions through virtual torus localization. If $\mathsf{\Gamma}$ is an abelian group the partition function is expressed as a combinatorial series over arrays of $\mathsf{\Gamma}$-coloured plane partitions, while if $\mathsf{\Gamma}$ 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 $\mathsf{\Gamma}=\mathbb{Z}_n$ 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 $\mathbb{C}^2/\,\mathsf{\Gamma}\times\mathbb{C}^2$ 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.