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

Abstract

Abstract We compute the topological partition function (twisted index) of N $$ \mathcal{N} $$ = 2 U(N) Chern-Simons theory with an adjoint chiral multiplet on Σ g × 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 λ (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 N $$ \mathcal{N} $$ = 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.

Keywords

• Topological Field Theories
• Bethe Ansatz
• Chern-Simons Theories