Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

TL;DR

This paper establishes a deep connection between Coulomb branch indices of class S theories and equivariant Verlinde formulas, revealing dualities and providing computational tools for complex gauge groups.

Contribution

It demonstrates the equivalence between Coulomb branch indices and equivariant Verlinde formulas, deriving new computational methods and exploring dualities for non-simply-connected gauge groups.

Findings

Proves the equivalence between Coulomb branch index and equivariant Verlinde formula.

Provides a recipe for computing indices with non-trivial 't Hooft fluxes.

Offers explicit checks for G=SU(2) and SO(3), and extends to SU(N) and PSU(N).

Abstract

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.