New Recipes for Brownian Loop Soups

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 $c = 2 λ$, where $λ> 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i βN(z)}$ 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 ${\fracλ{10}}(1 - \cos β)$. Here we generalize this procedure by assigning a more general random value to each loop. The operator $e^{i βN(z)}$ remains primary with conformal dimension $\frac λ{10}(1 - ϕ(β))$, where $ϕ(β)$ 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 $λ, β_i, z_i$, and on the characteristic function $ϕ(β)$. 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.