{
  "id": "math/9803026",
  "version": "v2",
  "published": "1998-03-09T18:26:07.000Z",
  "updated": "2001-01-23T22:34:48.000Z",
  "title": "On the quantum cohomology of a symmetric product of an algebraic curve",
  "authors": [
    "Aaron Bertram",
    "Michael Thaddeus"
  ],
  "comment": "28 pages; LaTeX2e, with packages amsfonts, latexsym, eepic",
  "categories": [
    "math.AG",
    "math.DG",
    "math.SG"
  ],
  "abstract": "The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [3/4 g, g-1). Otherwise, they still give a formula modulo third order terms. Explicit generators and relations are also given unless d is in [4/5 g - 3/5, g-1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically a formula of Harris-Tu for the Chern numbers of determinantal varieties. The case d = g-1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with Givental's work is also explained.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-01-23T22:34:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H99"
    ],
    "keywords": [
      "algebraic curve",
      "formula modulo third order terms",
      "dth symmetric product",
      "aspinwall-morrison formula enters",
      "little quantum cohomology"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......3026B"
    }
  }
}