Given a homomorphism \(\tau \) from a suitable finite group \({\mathsf {\Gamma }}\) to \(\textsf{SU}(4)\) with image \({\mathsf {\Gamma }}^\tau \) , we construct a cohomological gauge theory on a non-commutative resolution of the quotient singularity \(\mathbbm {C}^4/{\mathsf {\Gamma }}^\tau \) 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 \([\mathbbm {C}^4/\,{\mathsf {\Gamma }}^\tau ]\) . 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 \({\mathsf {\Gamma }}=\mathbbm {Z}_n\) is a finite abelian subgroup of \(\textsf{SL}(2,\mathbbm {C})\) , we exhibit the reduction of Donaldson–Thomas theory on the toric Calabi–Yau four-orbifold \(\mathbbm {C}^2/\,{\mathsf {\Gamma }}\times \mathbbm {C}^2\) 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 correspondence to derive a closed formula for the partition function on any polyhedral singularity.
