New Recipes for Brownian Loop Soups

TL;DR

This paper introduces a broad class of conformal primary operators in Brownian loop soups, computes their correlation functions analytically, and explores their conformal properties and primary operator spectrum.

Contribution

It generalizes the construction of primary operators in Brownian loop soups by assigning arbitrary random values, and derives explicit correlation functions depending on the characteristic function.

Findings

Correlation functions depend analytically on parameters

Operators satisfy conformal Ward identities

Four-point functions reveal additional primary operators

Abstract

We define a large new class of conformal primary operators in the ensemble of Brownian loops in two dimensions known as the ``Brownian loop soup,'' and compute their correlation functions analytically and in closed form. The loop soup is a conformally invariant statistical ensemble with central charge $c = 2 \lambda$, where $\lambda > 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i \beta N(z)}$ as primary operators. Each Brownian loop was assigned $\pm 1$ randomly, and $N(z)$ was defined to be the sum of these numbers over all loops that encircle the point $z$. These exponential operators then have conformal dimension ${\frac{\lambda}{10}}(1 - \cos \beta)$. Here we generalize this procedure by assigning a more general random value to each loop. The operator $e^{i \beta N(z)}$ remains primary with conformal dimension $\frac {\lambda}{10}(1 - \phi(\beta))$, where $\phi(\beta)$ is the characteristic function of the probability distribution used to assign random values to each loop. Using recent results we compute in closed form the exact two-point functions in the upper half-plane and four-point functions in the full plane of this very general class of operators. These correlation functions depend analytically on the parameters $\lambda, \beta_i, z_i$, and on the characteristic function $\phi(\beta)$. They satisfy the conformal Ward identities and are crossing symmetric. As in previous work, the conformal block expansion of the four-point function reveals the existence of additional and as-yet uncharacterized conformal primary operators.