Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

Ugo Bruzzo, Mattia Pedrini, Francesco Sala*, Richard J. Szabo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)
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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 ℝ4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak-1 toric singularity ℂ2/ℤk, which is a projective toric orbifold Xk such that Xk{set minus}Xk is a ℤk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective 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 Xk 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 Xk. 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 Seiberg-Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg-Witten curve with characteristics related to the nontrivial holonomies.

Original languageEnglish
Pages (from-to)1175-1308
Number of pages134
JournalAdvances in Mathematics
Early online date9 Dec 2015
Publication statusPublished - 22 Jan 2016


  • ALE spaces
  • Blowup formulas
  • Framed sheaves
  • Partition functions
  • Stacks
  • Supersymmetric gauge theories

ASJC Scopus subject areas

  • Mathematics(all)


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