Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

Sergei Gukov California Institute of Technology, Max-Planck-Institute for Mathematics Du Pei California Institute of Technology Wenbin Yan Harvard University CMSA, California Institute of Technology Ke Ye California Institute of Technology

Publications of CMSA of Harvard mathscidoc:1702.38067

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory T[Σ,G] on L(k,1)×S1, the other is the LG "equivariant Verlinde formula", or equivalently partition function of LGℂ complex Chern-Simons theory on Σ×S1. 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 LG. When G is not simply-connected, we provide a recipe of computing the index of T[Σ,G] as summation over indices of T[Σ,G̃ ] with non-trivial background 't Hooft fluxes, where G̃ is the simply-connected group with the same Lie algebra. 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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@inproceedings{sergeiequivariant,
  title={Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality},
  author={Sergei Gukov, Du Pei, Wenbin Yan, and Ke Ye},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208001222550080293},
}
Sergei Gukov, Du Pei, Wenbin Yan, and Ke Ye. Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208001222550080293.
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