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

Wan Keng Cheong, Ngau Lam

TL;DR

This work develops the Gaudin model for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ by constructing the Gaudin algebra $\mathfrak B_{m|n}(\underline{\boldsymbol z})$ from the Feigin–Frenkel center and Berezinians, and studies its action on the singular space $M^{\mathrm{sing}}$ of an $\ell$-fold tensor product of polynomial modules. It proves that $M^{\mathrm{sing}}$ is cyclic and that $\mathfrak B_{m|n}(\underline{\boldsymbol z})_{M^{\mathrm{sing}}}$ is a Frobenius algebra, diagonalizable with simple spectrum for generic site configurations, and that its eigenbasis can be described via Fuchsian differential operators with polynomial kernels, providing a reformulation of the Bethe ansatz in the super setting. By leveraging odd reflections, truncation functors, and a tensor Shapovalov form, the paper extends the Mukhin–Tarasov–Varchenko correspondence to $\mathfrak{gl}_{m|n}$, linking eigenvalues to differential operator data and suggesting a super Langlands-type perspective for Gaudin models. The results indicate a robust structure where Bethe-type eigenvectors, when available for related algebras, extend to give complete eigenbases for the super Gaudin algebra, with central roles played by the Feigin–Frenkel center and its polynomial generators.

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 $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$.

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

TL;DR

This work develops the Gaudin model for the general linear Lie superalgebra by constructing the Gaudin algebra from the Feigin–Frenkel center and Berezinians, and studies its action on the singular space of an -fold tensor product of polynomial modules. It proves that is cyclic and that is a Frobenius algebra, diagonalizable with simple spectrum for generic site configurations, and that its eigenbasis can be described via Fuchsian differential operators with polynomial kernels, providing a reformulation of the Bethe ansatz in the super setting. By leveraging odd reflections, truncation functors, and a tensor Shapovalov form, the paper extends the Mukhin–Tarasov–Varchenko correspondence to , linking eigenvalues to differential operator data and suggesting a super Langlands-type perspective for Gaudin models. The results indicate a robust structure where Bethe-type eigenvectors, when available for related algebras, extend to give complete eigenbases for the super Gaudin algebra, with central roles played by the Feigin–Frenkel center and its polynomial generators.

Abstract

Let be the Gaudin algebra of the general linear Lie superalgebra with respect to a sequence of pairwise distinct complex numbers, and let be any -fold tensor product of irreducible polynomial modules over . We show that the singular space of is a cyclic -module and the Gaudin algebra of is a Frobenius algebra. We also show that is diagonalizable with a simple spectrum for a generic 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 .
Paper Structure (18 sections, 32 theorems, 100 equations)

This paper contains 18 sections, 32 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The general linear Lie superalgebra ${\mathfrak{gl}}_{m|n}$
  4. Berezinians
  5. Pseudo-differential operators
  6. The Gaudin model of ${\mathfrak{gl}}_{m|n}$
  7. The Feigin--Frenkel center
  8. The Gaudin algebra of ${\mathfrak{gl}}_{m|n}$
  9. Polynomial modules
  10. Tensor products of polynomial modules
  11. Main results
  12. Frobenius algebras
  13. The Gaudin algebra of ${\mathfrak{gl}}_{m}$
  14. Proofs of the main results
  15. The completeness of the Bethe ansatz and the Feigin--Frenkel center
  16. ...and 3 more sections

Key Result

Theorem 1.1

$M^{\rm sing}$ is a cyclic ${\mathfrak B}_{m|n}(\underline{\boldsymbol z})$-module, and ${\mathfrak B}_{m|n}(\underline{\boldsymbol z})_{M^{\rm sing}}$ is a Frobenius algebra for any sequence $\underline{\boldsymbol z} \in {\mathbb C}^\ell$ of pairwise distinct complex numbers.

Theorems & Definitions (51)

  • Theorem 1.1: Theorem \ref{['cyc+frob']}
  • Theorem 1.2: Theorem \ref{['diag']}
  • Proposition 2.1
  • Proposition 2.2: HM
  • Proposition 2.3: HM
  • Proposition 2.4: CFR
  • Proposition 2.5: MR
  • Proposition 3.1
  • Proposition 3.2: MR
  • Proposition 3.3
  • ...and 41 more