The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

TL;DR

This paper proves the completeness of the Bethe ansatz for the Gaudin model associated with the general linear Lie superalgebra, showing diagonalizability and explicit eigenbasis construction for the model's algebraic structures.

Contribution

It establishes the cyclicity, Frobenius property, and diagonalizability of the Gaudin algebra's action on singular spaces, extending Bethe ansatz completeness to superalgebra contexts.

Findings

Singular space is a cyclic module over the Gaudin algebra.

The Gaudin algebra on the singular space is Frobenius.

Eigenbasis and eigenvalues are described via Fuchsian differential operators.

Abstract

Let $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$ be the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ with respect to a sequence $\underline{\boldsymbol{z}} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $M$ be any $\ell$-fold tensor product of irreducible polynomial modules over $\mathfrak{gl}_{m|n}$. We show that the singular space $M^{\rm sing}$ of $M$ is a cyclic $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$-module and the Gaudin algebra $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ of $M^{\rm sing}$ is a Frobenius algebra. We also show that $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ is diagonalizable with a simple spectrum for a generic $\underline{\boldsymbol{z}}$ and give a description of an eigenbasis and its corresponding eigenvalues in terms of the Fuchsian differential operators with polynomial kernels. This may be interpreted as the completeness of a reformulation of the Bethe ansatz for $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$.