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

Richard J. Szabo, Michelangelo Tirelli

Publication date: 30 January 2023

Abstract: We study rank

r

cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of

m a t h b b C^{4}

by a finite abelian subgroup

m a t h s f G a m m a

of

m a t h s f S U (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

m a t h b b C^{4} / m a t h s f G a m m a

and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over

r

-vectors of

m a t h s f G a m m a

-coloured solid partitions. When the

m a t h s f G a m m a

-action fixes an affine line in

m a t h b b C^{4}

, we exhibit the dimensional reduction to rank

r

Donaldson-Thomas theory on the toric Kahler three-orbifold

m a t h b b C^{3} / m a t h s f G a m m a

. 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

m a t h b b C^{2} / m a t h b b Z_{n} i m e s m a t h b b C^{2}

and

m a t h b b C^{3} / (m a t h b b Z_{2} i m e s m a t h b b Z_{2}) i m e s m a t h b b C

, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.

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