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 S theory $T[\Sigma,G]$ on $L(k,1) \times S^1$, the other is the $^LG$ "equivariant Verlinde formula", or equivalently partition function of $^LG_{\mathbb{C}}$ 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 $^LG$. When $G$ is not simply-connected, we provide a recipe of computing the index of $T[\Sigma,G]$ as summation over indices of $T[\Sigma,\tilde{G}]$ with non-trivial background 't Hooft fluxes, where $\tilde{G}$ 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.