The Gaudin model and the super Bethe ansatz for unitarizable modules over the general linear Lie (super)algebra
Year: 2026
Type: preprint
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 $\mathbb{L}^\text{sing}_ξ$.
Source: arXiv (Cornell University)
Authors: Cao, Bintao, Cheong, Wan Keng, Lam, Ngau
Institutions: Yunnan University, National Cheng Kung University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6681542643
OpenAlex ID: W7213248844