Communications in Mathematical Physics

, Volume 357, Issue 3, pp 1215–1251 | Cite as

Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality

  • Sergei Gukov
  • Du Pei
  • Wenbin Yan
  • Ke Ye


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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Copyright information

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

Authors and Affiliations

  1. 1.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Center for Quantum Geometry of Moduli Spaces, Department of MathematicsAarhus UniversityAarhusDenmark
  4. 4.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  5. 5.Center for Mathematical Sciences and ApplicationsHarvard UniversityCambridgeUSA
  6. 6.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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