The Gaudin model and the super Bethe ansatz for unitarizable modules over the general linear Lie (super)algebra

Abstract

Let $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ be the Gaudin algebra for the general linear Lie (super)algebra $\mathfrak{g}$ with respect to a sequence ${\bf z} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $\mathbb{L}$ be any $\ell$-fold tensor product of (infinite-dimensional) unitarizable highest weight $\mathfrak{g}$-modules. Fix a singular weight $ξ$ of $\mathbb{L}$. We show that the singular weight space $\mathbb{L}^\text{sing}_ξ$ is a cyclic $\mathcal{A}^{\bf z}_{\mathfrak{g}}$-module, and that the Gaudin algebra $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_ξ}$ for $\mathbb{L}^\text{sing}_ξ$ is a Frobenius algebra. We also show that $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_ξ}$ is diagonalizable with a simple spectrum for a generic ${\bf z}$. Furthermore, we establish a super version of the Bethe ansatz for $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ on $\mathbb{L}^\text{sing}_ξ$.