Abstract
We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on ℝ^{4} and with twodimensional conformal field theory. We construct a stacky compactification of the minimal resolution X_{k} of the A_{k1} toric singularity ℂ^{2}/ℤ_{k}, which is a projective toric orbifold X_{k} such that X_{k}{set minus}X_{k} is a ℤ_{k}gerbe. We construct moduli spaces of torsion free sheaves on X_{k} which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasiprojective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on X_{k} with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of X_{k}. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the SeibergWitten curves, and in some cases also by suitable blowup equations involving Riemann thetafunctions on the SeibergWitten curve with characteristics related to the nontrivial holonomies.
Original language  English 

Pages (fromto)  11751308 
Number of pages  134 
Journal  Advances in Mathematics 
Volume  288 
Early online date  9 Dec 2015 
DOIs  
Publication status  Published  22 Jan 2016 
Keywords
 ALE spaces
 Blowup formulas
 Framed sheaves
 Partition functions
 Stacks
 Supersymmetric gauge theories
ASJC Scopus subject areas
 General Mathematics
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Richard Joseph Szabo
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)