{
  "id": "2111.01106",
  "version": "v3",
  "published": "2021-11-01T17:20:53.000Z",
  "updated": "2022-03-22T13:10:21.000Z",
  "title": "Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model",
  "authors": [
    "Linnea Grans-Samuelsson",
    "Rongvoram Nivesvivat",
    "Jesper Lykke Jacobsen",
    "Sylvain Ribault",
    "Hubert Saleur"
  ],
  "comment": "49 pages, v3: improved explanations on a few points",
  "categories": [
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\\in\\mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-03-22T13:10:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conformal bootstrap",
      "global symmetry",
      "two-dimensional",
      "representation theory",
      "numerically bootstrap arbitrary four-point functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}