{
  "id": "1406.5158",
  "version": "v2",
  "published": "2014-06-19T19:15:41.000Z",
  "updated": "2014-10-05T18:25:43.000Z",
  "title": "Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$",
  "authors": [
    "Iana I. Anguelova"
  ],
  "comment": "23 pages. References added, some typos corrected",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA",
    "math.RT"
  ],
  "abstract": "We construct the bosonization of the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ as a direct sum of irreducible highest weight modules for the Heisenberg algebra $\\mathcal{H}_{\\mathbb{Z}}$, and thus we show that under the Heisenberg $\\mathcal{H}_{\\mathbb{Z}}$ action the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ of the single neutral fermion is isomorphic to the Fock space $\\mathit{F^{\\otimes 1}}$ of a pair of charged free fermions, thereby constructing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on $\\mathit{F^{\\otimes \\frac{1}{2}}}$. We construct a family of 2-point-local Virasoro fields with central charge $-2+12\\lambda -12\\lambda^2, \\ \\lambda\\in \\mathbb{C}$, on the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$. We construct a $W_{1+\\infty}$ representation on $\\mathit{F^{\\otimes \\frac{1}{2}}}$ and show that under the $W_{1+\\infty}$ action $\\mathit{F^{\\otimes \\frac{1}{2}}}$ is again isomorphic to $\\mathit{F^{\\otimes 1}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-19T19:15:41.000Z",
      "abstract": "We construct the bosonization of the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ as a direct sum of irreducible highest weight modules for the Heisenberg algebra $\\mathcal{H}_{\\mathbb{Z}}$, thereby constructing the boson-fermion correspondence of type D-A. We show that under the Heisenberg $\\mathcal{H}_{\\mathbb{Z}}$ action the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$ of the single neutral fermion is isomorphic to the Fock space $\\mathit{F^{\\otimes 1}}$ of a pair of charged free fermions. By utilizing both the Heisenberg and the Virasoro gradings we obtain the Jacobi identity equating the two graded dimension formulas for $\\mathit{F^{\\otimes \\frac{1}{2}}}$. We construct a family of 2-point-local Virasoro fields with central charge $-2+12\\lambda -12\\lambda^2, \\ \\lambda\\in \\mathbb{C}$, on the Fock space $\\mathit{F^{\\otimes \\frac{1}{2}}}$. We construct a $W_{1+\\infty}$ representation on $\\mathit{F^{\\otimes \\frac{1}{2}}}$ and show that under the $W_{1+\\infty}$ action $\\mathit{F^{\\otimes \\frac{1}{2}}}$ is again isomorphic to $\\mathit{F^{\\otimes 1}}$.",
      "comment": "23 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-05T18:25:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81R10",
      "17B68",
      "17B69"
    ],
    "keywords": [
      "fock space",
      "multi-local virasoro representations",
      "boson-fermion correspondence",
      "type d-a",
      "single neutral fermion"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.4901557",
      "journal": "Journal of Mathematical Physics",
      "year": 2014,
      "month": "Nov",
      "volume": 55,
      "number": 11,
      "pages": 111704
    },
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1301612,
      "adsabs": "2014JMP....55k1704A"
    }
  }
}