Letters in Mathematical Physics

, Volume 105, Issue 3, pp 401–445 | Cite as

\({\mathcal{N} = 2}\) Quiver Gauge Theories on A-type ALE Spaces



We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in \({\mathcal{N} = 2}\) gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xk of the Ak-1 toric singularity \({\mathbb{C}^2/\mathbb{Z}_k}\) , in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion-free sheaves on a suitable orbifold compactification of Xk. We extend these computations to generic \({\mathcal{N} = 2}\) superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg–Witten geometry.


stacks framed sheaves ALE spaces supersymmetric gauge theories partition functions blowup formulas 

Mathematics Subject Classification

14D20 14D21 14J80 81T13 81T60 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Ugo Bruzzo
    • 1
    • 2
  • Francesco Sala
    • 3
  • Richard J. Szabo
    • 4
    • 5
    • 6
  1. 1.Scuola Internazionale Superiore di Studi Avanzati (SISSA)TriesteItaly
  2. 2.Istituto Nazionale di Fisica Nucleare, Sezione di TriesteTriesteItaly
  3. 3.Department of MathematicsThe University of Western Ontario, Middlesex CollegeLondonCanada
  4. 4.Department of MathematicsHeriot-Watt UniversityRiccartonUK
  5. 5.Maxwell Institute for Mathematical SciencesEdinburghUK
  6. 6.The Tait InstituteEdinburghUK

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