Scalar conformal primary fields in the Brownian loop soup

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

Published 2021-09-24, updated 2022-02-01Version 2

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.