2023/01/30 by Richard J. Szabo, Szabo, Richard J., Michelangelo Tirelli +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2301.13069
We study rank r cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of ℂ4 by a finite abelian subgroup \mathsfΓ of SU(4), from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on ℂ4/\mathsfΓ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over r-vectors of \mathsfΓ-coloured solid partitions. When the \mathsfΓ-action fixes an affine line in ℂ4, we exhibit the dimensional reduction to rank r Donaldson-Thomas theory on the toric Kahler three-orbifold ℂ3/\mathsfΓ. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds ℂ2/ℤn×ℂ2 and ℂ3/(ℤ2×ℤ2)×ℂ, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.