{
  "id": "2109.12116",
  "version": "v2",
  "published": "2021-09-24T18:00:00.000Z",
  "updated": "2022-02-01T11:46:17.000Z",
  "title": "Scalar conformal primary fields in the Brownian loop soup",
  "authors": [
    "Federico Camia",
    "Valentino F. Foit",
    "Alberto Gandolfi",
    "Matthew Kleban"
  ],
  "comment": "40 pages, 2 figures, clarified the relation to the scaling limit of critical percolation, corrected definition (2.3) and equations depending on it, corrected proof of Lemma 2.2, added Lemma A.2",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "hep-th",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $\\lambda>0$, with central charge $c=2 \\lambda$. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary \"layering vertex operators\" $\\mathcal{O}_{\\beta}$ with dimensions $(\\Delta, \\Delta)$, with $\\Delta = \\frac{\\lambda}{10}(1-\\cos\\beta)$, that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions $(\\Delta+ k/3, \\Delta + k'/3)$, for all non-negative integers $k, k'$ satisfying $|k-k'| = 0$ mod 3. In this paper we introduce the edge counting field $\\mathcal E(z)$ that counts the number of loop boundaries that pass close to the point $z$. We rigorously prove that the $n$-point functions of $\\mathcal E$ are well defined and behave as expected for a conformal primary field with dimensions $(1/3, 1/3)$. We analytically compute the four-point function $\\langle \\mathcal{O}_{\\beta}(z_1) \\mathcal{O}_{-\\beta}(z_2) \\mathcal{E}(z_3) \\mathcal{E}(z_4) \\rangle$ and analyze its conformal block expansion. The operator product expansions of $\\mathcal{E} \\times \\mathcal{E}$ and $\\mathcal{E} \\times \\mathcal{O}_{\\beta}$ produce higher-order edge operators with \"charge\" $\\beta$ and dimensions $(\\Delta + k/3, \\Delta + k/3)$. Hence, we have explicitly identified all scalar primary operators among the new set mentioned above. We also re-compute the central charge by an independent method based on the operator product expansion and find agreement with previous methods.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-01T11:46:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scalar conformal primary fields",
      "brownian loop soup",
      "conformal block expansion",
      "operator product expansion",
      "four-point function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}