Tetrahedron Instantons on Orbifolds

Given a homomorphism τ from a finite group Γ to SU(4) with image Γ^τ, we construct a cohomological gauge theory on a noncommutative resolution of the quotient singularity C⁴/Γ^τ whose BRST fixed points are 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 [C⁴/\,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 Γ is an abelian group the partition function is expressed as a combinatorial series over arrays of Γ-coloured plane partitions, while if 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 Γ=Zₙ is a finite abelian subgroup of SL(2,C), we exhibit the reduction of Donaldson-Thomas theory on the toric Calabi-Yau four-orbifold C²/\,Γ² 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.