Exact Correlation Functions in the Brownian Loop Soup

TL;DR

This paper derives exact, closed-form expressions for four- and two-point correlation functions of layering vertex operators in the conformally invariant Brownian Loop Soup, revealing new primary operators and their conformal structure.

Contribution

It provides the first exact formulas for these correlation functions in the Brownian Loop Soup, including analysis of conformal blocks and discovery of new primary operators.

Findings

Exact four-point correlation function in the plane.

Exact two-point correlation function in the upper half-plane.

Identification of new conformal primary operators.

Abstract

We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.