Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

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 ${}^{L}G$ "equivariant Verlinde formula", or equivalently partition function of ${}^L{G}_{\mathbb{C}}$ 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 ${}^{L}G$. 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.

40 pages, 7 figures, Mathematica Notebook attached; v2: misprints corrected; v3: Acknowledgement added, corrections made based on the journal version

References in corpus (6)

• S-duality in N=2 supersymmetric gauge theories
• Tinkertoys for Gaiotto Duality
• From Rigid Supersymmetry to Twisted Holomorphic Theories
• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• The Verlinde formula for Higgs bundles
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Cited by in corpus (11)

• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• A slow review of the AGT correspondence
• The Verlinde formula for Higgs bundles
• Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra
• Argyres-Douglas matter and S-duality: Part II
• 3d TQFTs from Argyres-Douglas theories
• Mirror symmetry with branes by equivariant Verlinde formulae
• Equivariant U(N) Verlinde algebra from Bethe/Gauge correspondence
• Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories