{
  "id": "1708.05105",
  "version": "v1",
  "published": "2017-08-16T23:26:30.000Z",
  "updated": "2017-08-16T23:26:30.000Z",
  "title": "Crystals and monodromy of Bethe vectors",
  "authors": [
    "Iva Halacheva",
    "Joel Kamnitzer",
    "Leonid Rybnikov",
    "Alex Weekes"
  ],
  "comment": "64 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-16T23:26:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bethe vectors",
      "tensor product multiplicity space",
      "eigenvectors",
      "internal cactus group action",
      "deligne-mumford moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}