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Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality

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Abstract

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the “Coulomb branch index” of the class \({{\mathcal S}}\) theory \({T[\Sigma,G]}\) on \({L(k,1) \times S^1}\), the other is the \({^L G}\) “equivariant Verlinde formula”, or equivalently partition function of \({^L G_{\mathbb{C}}}\) complex Chern–Simons theory on \({\Sigma\times S^1}\). We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual \({^L G}\). When G is not simply-connected, we provide a recipe of computing the index of \({T[\Sigma,G]}\) as summation over the indices of \({T[\Sigma,\tilde{G}]}\) with non-trivial background ’t Hooft fluxes, where \({\tilde{G}}\) is the universal cover of G. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for \({G=SU(2)}\) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres–Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients.

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Authors and Affiliations

  1. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    Sergei Gukov, Du Pei, Wenbin Yan & Ke Ye

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Sergei Gukov

  3. Center for Quantum Geometry of Moduli Spaces, Department of Mathematics, Aarhus University, 8000, Aarhus, Denmark

    Du Pei

  4. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Wenbin Yan

  5. Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02138, USA

    Wenbin Yan

  6. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Wenbin Yan

Authors
  1. Sergei Gukov
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  2. Du Pei
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  3. Wenbin Yan
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  4. Ke Ye
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Correspondence to Ke Ye.

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Communicated by X. Yin

Wenbin Yan—Primary affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing, China.

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Gukov, S., Pei, D., Yan, W. et al. Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality. Commun. Math. Phys. 357, 1215–1251 (2018). https://doi.org/10.1007/s00220-017-3074-8

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  • DOI: https://doi.org/10.1007/s00220-017-3074-8

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