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Communications in Mathematical Physics

, Volume 357, Issue 2, pp 519–567 | Cite as

Quantum Geometry and Quiver Gauge Theories

  • Nikita Nekrasov
  • Vasily Pestun
  • Samson Shatashvili
Article

Abstract

We study macroscopically two dimensional \({\mathcal{N}=(2,2)}\) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product \({\mathbb{T}^{d} \times \mathbb{R}^{2}_{\epsilon}}\) of a d-dimensional torus and a two dimensional cigar with \({\Omega}\)-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories and the Yangian \({\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})}\), quantum affine algebra \({\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})}\), or the quantum elliptic algebra \({\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})}\) associated to Kac–Moody algebra \({\mathfrak{g}_{\Gamma}}\) for quiver \({\Gamma}\).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Nikita Nekrasov
    • 1
  • Vasily Pestun
    • 2
  • Samson Shatashvili
    • 3
    • 4
  1. 1.Simons Center for Geometry and PhysicsStony BrookUSA
  2. 2.Institut des Hautes Études ScientifiqueBures-sur-YvetteFrance
  3. 3.The Hamilton Mathematics Institute, School of MathematicsTrinity College DublinDublinIreland
  4. 4.Chaire Louis MichelInstitut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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