Scalar conformal primary fields in the Brownian loop soup

TL;DR

This paper introduces and rigorously analyzes the edge counting field in the Brownian loop soup, identifying new scalar primary operators and computing their correlation functions, thereby advancing understanding of the model's conformal structure.

Contribution

It introduces the edge counting field, proves its conformal primary nature, and explicitly identifies new scalar primary operators in the Brownian loop soup.

Findings

Edge counting field $ extcal{E}$ is a conformal primary with dimensions (1/3, 1/3).

Explicit four-point function involving $ extcal{E}$ and layering operators computed.

Higher-order edge operators with specific charge and dimensions identified.

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.