Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

TL;DR

The paper identifies and substantiates a deep equivalence between two a priori different 2d TQFT constructions: the equivariant Verlinde algebra of a gauge group $G$ and the Coulomb branch index of the class S theory $T[\Sigma,G]$ on $L(k,1)\times S^1$. By leveraging M-theory realizations, Langlands duality, and flux-sum prescriptions, it shows that the Verlinde data for $G_k$ on $\Sigma$ correspond to the Coulomb branch index of $T[\Sigma,^L G]$ with $k$-dependent flux sectors, and it generalizes to non-simply connected groups via the appropriate flux bookkeeping. The authors explicitly verify the correspondence for $SU(2)$ and $SO(3)$ and then derive the $SU(N)$ (and $PSU(N)$) equivariant Verlinde algebras through generalized Argyres-Seiberg duality, including a detailed $SU(3)$ analysis using the $T_3$ theory and its $E_6$ symmetry. They also connect the quantum algebra to Hitchin moduli space geometry, offering fixed-point localization interpretations for fusion coefficients and enabling computation of higher-rank Verlinde data. The work provides concrete tools and a unifying framework for computing and interpreting Verlinde coefficients via 4d SCFT dualities and Lens space indices, with a Mathematica notebook supplied for practical $SU(3)$ calculations.

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 $T[Σ,G]$ 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 $Σ\times S^1$. 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 $T[Σ,G]$ as summation over indices of $T[Σ,\tilde{G}]$ 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.

Figures (7)

• Figure 1: Illustration of Argyres-Seiberg duality. (a) The theory A, which is an $SU(3)$ superconformal gauge theory with six hypermultiplets, with the $SU(3)_a \times U(1)_a \times SU(3)_b \times U(1)_b$ subgroup of the global $U(6)$ flavor symmetry. (b) The theory B, obtained by gauging an $SU(2)$ subgroup of the $E_6$ symmetry of $T_3$. Note in the geometric realization the cylinder connecting both sides has a regular puncture $R$ on the left and an irregular puncture $IR$ on the right.
• Figure 2: Illustration of geometric realization of Argyres-Seiberg duality for $T_3$ theory. The dots represent simple punctures while circles are maximal punctures. (a) The theory A, which is an $SU(3)$ superconformal gauge theory with six hypermultiplets, is pictured as two spheres connected by a long tube. Each of them has two maximal and one simple punctures. (b) The theory B, which is obtained by gauging an $SU(2)$ subgroup of the flavor symmetry of the theory $T_3$. This gauge group connects a regular puncture and an irregular puncture.
• Figure 3: The Weyl alcove for the choice of holonomy variables at level $k=3$. The red markers represent the allowed points. The coordinates beside each point denote the corresponding highest weight representation. The transformation between flavor holonomies and highest weight is given by \ref{['highestWeight']}.
• Figure 4: The illustration of the nilpotent cone in ${{\mathcal{M}}}_H(\Sigma_{0,3}, SU(3))$. Here ${\mathcal{M}}$ is the base ${\mathbb{C}}{\mathbf{P}}^1$, $D_{1,2,3}$ consist of downward Morse flows from $P_{1,2,3}$ to the base, while $D_{4,5,6}$ include the flows from $P_{4,5,6}$ to $P_{1,2,3}$.
• Figure 5: The affine $\widehat{E}_6$ extended Dynkin diagram. The Dynkin label gives the multiplicity of each node in the decomposition of the null vector.