Boundary operators in the Brownian loop soup

TL;DR

The work identifies an infinite family of boundary primaries in the subcritical Brownian loop soup by performing a conformal block decomposition of the upper-half-plane two-point function of layering operators, showing that their boundary dimensions are non-negative integers and that even-dimensional cases correspond to inserting multiple outer loop boundaries on the real axis. It derives both s- and t-channel expansions to produce bulk-boundary OPE data, revealing boundary operators such as $Y(x)$ and $e^{(2k)}(x)$, and provides a probabilistic interpretation in terms of Brownian bubble and loop measures, notably $\mu^{\text{bub}}_{\mathbb{H}}$ and $\mu^{\text{loop}}_{\mathbb{H}}$. To address inconsistencies with the existing edge operator construction, the paper introduces a refined definition of the edge operator on the half-plane that yields nonzero one-point functions and a coherent bulk-boundary OPE, including a $\Delta=\tfrac{2}{3}$ scaling for the edge field. The results illuminate a path toward a full 2D CFT description of the Brownian loop soup with $c\le 1$, discuss potential symmetries implied by integer-dimension primaries, and lay out future work on correlation functions of $e^{(2k)}(x)$ and the role of conformal invariance in this probabilistic setting.

Abstract

We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper half-plane being separated by Brownian loops. The resulting boundary operators are primary operators in a 2D CFT with central charge $c\leq1$ and have conformal dimensions that are non-negative integers. By comparing the above-mentioned conformal block expansion with probabilities in the Brownian loop soup, we provide a physical interpretation of the boundary operators of even dimensions as operators that insert multiple outer boundaries of Brownian loops at points on the real axis.