TY - UNPB

T1 - Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds

AU - Szabo, Richard J.

AU - Tirelli, Michelangelo

N1 - 66 pages; v2: typos fixed, comments added; v3: exposition improved, appendix on generalised ADHM construction added; Final version to appear in Advances in Theoretical and Mathematical Physics

PY - 2023/1/30

Y1 - 2023/1/30

N2 - We study rank r cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of C4 by a finite abelian subgroup Γ 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/Γ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over r-vectors of Γ-coloured solid partitions. When the Γ-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/Γ. 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.

AB - We study rank r cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of C4 by a finite abelian subgroup Γ 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/Γ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over r-vectors of Γ-coloured solid partitions. When the Γ-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/Γ. 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.

KW - hep-th

KW - math-ph

KW - math.AG

KW - math.MP

KW - math.QA

M3 - Preprint

BT - Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds

PB - arXiv

ER -