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

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.