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

Abstract

Let ${\mathfrak{B}}_{m|n}(\underline{\mathbf{z}})$ be the Gaudin algebra of the general linear Lie superalgebra ${\mathfrak{gl}}_{m|n}$ with respect to a sequence $\underline{\mathbf{z}}\in {\mathbb{C}}^{\ell }$ 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 $M^{\mathrm{sing}}$ of $M$ is a cyclic ${\mathfrak{B}}_{m|n}(\underline{\mathbf{z}})$-module and the Gaudin algebra ${\mathfrak{B}}_{m|n}(\underline{\mathbf{z}}{)}_{M^{\mathrm{sing}}}$ of $M^{\mathrm{sing}}$ is a Frobenius algebra. We also show that ${\mathfrak{B}}_{m|n}(\underline{\mathbf{z}}{)}_{M^{\mathrm{sing}}}$ is diagonalizable with a simple spectrum for a generic $\underline{\mathbf{z}}$ 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 ${\mathfrak{B}}_{m|n}(\underline{\mathbf{z}}{)}_{M^{\mathrm{sing}}}$.