Equivariant U(N) Verlinde algebra from Bethe/Gauge correspondence

Abstract

We compute the topological partition function (twisted index) of $\mathcal{N}=2$ $U(N)$ Chern-Simons theory with an adjoint chiral multiplet on ${\Sigma }_g\times S^1$. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by $U(1)$ equivariant parameter $t$. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials $P_{\lambda }(x_B,t)$ with a specialization by Bethe roots $x_B$ of the $q$-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional $\mathcal{N}=2$ theories for $SU(2)$ and $SU(3)$ with lower levels. In $SU(2)$ case we also present more direct computation based on Jeffrey-Kirwan residue operation.

45 pages, 4 figures

References in corpus (6)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Level/rank Duality and Chern-Simons-Matter Theories
• Comments on twisted indices in 3d supersymmetric gauge theories
• Bethe/Gauge correspondence on curved spaces
• G/G gauged WZW model and Bethe Ansatz for the phase model
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

Cited by in corpus (3)

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• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories