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

Abstract

Let ${\mathcal{A}}_{\mathfrak{g}}^{\mathbf{z}}$ be the Gaudin algebra for the general linear Lie (super)algebra $\mathfrak{g}$ with respect to a sequence $\mathbf{z}\in {\mathbb{C}}^{\ell }$ 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 $\xi$ of $\mathbb{L}$. We show that the singular weight space ${\mathbb{L}}_{\xi }^{\text{sing}}$ is a cyclic ${\mathcal{A}}_{\mathfrak{g}}^{\mathbf{z}}$-module, and that the Gaudin algebra $({\mathcal{A}}_{\mathfrak{g}}^{\mathbf{z}}{)}_{{\mathbb{L}}_{\xi }^{\text{sing}}}$ for ${\mathbb{L}}_{\xi }^{\text{sing}}$ is a Frobenius algebra. We also show that $({\mathcal{A}}_{\mathfrak{g}}^{\mathbf{z}}{)}_{{\mathbb{L}}_{\xi }^{\text{sing}}}$ is diagonalizable with a simple spectrum for a generic $\mathbf{z}$. Furthermore, we establish a super version of the Bethe ansatz for ${\mathcal{A}}_{\mathfrak{g}}^{\mathbf{z}}$ on ${\mathbb{L}}_{\xi }^{\text{sing}}$.