TY - JOUR

T1 - Scalar Conformal Primary Fields in the Brownian Loop Soup

AU - Camia, Federico

AU - Foit, Valentino F.

AU - Gandolfi, Alberto

AU - Kleban, Matthew

N1 - Funding Information:
The work of M.K. is partially supported by the NSF through the grant PHY-1820814. No new data were created during the study.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-022-04611-7

DO - 10.1007/s00220-022-04611-7

M3 - Article

AN - SCOPUS:85144191101

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -