The Brownian loop soup stress-energy tensor

TL;DR

This paper explicitly constructs the stress-energy tensor within the Brownian loop soup, demonstrating its conformal properties, operator product expansion, and boundary behavior, thus advancing the understanding of conformal invariance in this stochastic model.

Contribution

It provides an explicit expression for the BLS stress-energy tensor and analyzes its properties, including OPEs and boundary operators, linking stochastic loops to conformal field theory.

Findings

The stress-energy tensor in BLS is explicitly constructed.

The operator product expansion with other operators is computed.

The boundary operator generates boundary deformations and relates to Brownian excursions.

Abstract

The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $\lambda>0$. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is $c = 2 \lambda$. In the case of domains with boundary we identify a boundary operator that has properties consistent with the boundary stress-energy tensor. We show that this operator generates local deformations of the boundary and that it is related to a boundary operator that induces a Brownian excursion starting or ending at its insertion point.