Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class SS theory T[Σ,G] on L(k,1)×S^1, the other is the LGLG "equivariant Verlinde formula", or equivalently partition function of LGCLGC complex Chern–Simons theory on Σ×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 LGLG. When G is not simply-connected, we provide a recipe of computing the index of T[Σ,G] as summation over the indices of T[Σ,G] with non-trivial background 't Hooft fluxes, where 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.