Tetrahedron instantons on orbifolds

Abstract Given a homomorphism $$\tau $$ τ from a suitable finite group $${\mathsf {\Gamma }}$$ Γ to $$\textsf{SU}(4)$$ SU ( 4 ) with image $${\mathsf {\Gamma }}^\tau $$ Γ τ , we construct a cohomological gauge theory on a non-commutative resolution of the quotient singularity $$\mathbbm {C}^4/{\mathsf {\Gamma }}^\tau $$ C 4 / Γ τ whose BRST fixed points are $${\mathsf {\Gamma }}$$ Γ -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 $$[\mathbbm {C}^4/\,{\mathsf {\Gamma }}^\tau ]$$ [ C 4 / Γ τ ] . We describe the generalized ADHM parametrization of the tetrahedron instanton moduli space and evaluate the orbifold partition functions through virtual torus localization. If $${\mathsf {\Gamma }}$$ Γ is an abelian group the partition function is expressed as a combinatorial series over arrays of $${\mathsf {\Gamma }}$$ Γ -coloured plane partitions, while if $${\mathsf {\Gamma }}$$ Γ 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 $${\mathsf {\Gamma }}=\mathbbm {Z}_n$$ Γ = Z n is a finite abelian subgroup of $$\textsf{SL}(2,\mathbbm {C})$$ SL ( 2 , C ) , we exhibit the reduction of Donaldson–Thomas theory on the toric Calabi–Yau four-orbifold $$\mathbbm {C}^2/\,{\mathsf {\Gamma }}\times \mathbbm {C}^2$$ C 2 / Γ × 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.