Scalar Conformal Primary Fields in the Brownian Loop Soup

Federico Camia, Valentino F. Foit, Alberto Gandolfi, Matthew Kleban

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The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity λ> 0 , with central charge c= 2 λ. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary “layering vertex operators” Oβ with dimensions (Δ , Δ) , with Δ=λ10(1-cosβ), 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 (Δ + k/ 3 , Δ + k/ 3) , for all non-negative integers k, k satisfying |k-k′|=0mod3. In this paper we introduce the edge counting field 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 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 〈 Oβ(z1) O-β(z2) E(z3) E(z4) 〉 and analyze its conformal block expansion. The operator product expansions of E× E and E× Oβ contain higher-order edge operators with “charge” β and dimensions (Δ + k/ 3 , Δ + 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.

Original languageEnglish (US)
Pages (from-to)977-1018
Number of pages42
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jun 2023

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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