{
  "id": "2501.00401",
  "version": "v1",
  "published": "2024-12-31T11:40:54.000Z",
  "updated": "2024-12-31T11:40:54.000Z",
  "title": "The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz",
  "authors": [
    "Wan Keng Cheong",
    "Ngau Lam"
  ],
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "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 $\\mathfrak{B}_{m|n}(\\underline{\\boldsymbol{z}})_{M^{\\rm sing}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-31T11:40:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general linear lie superalgebra",
      "bethe ansatz",
      "gaudin model",
      "completeness",
      "gaudin algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}