# New Recipes for Brownian Loop Soups

Research paper by **Valentino F. Foit, Matthew Kleban**

Indexed on: **07 Jul '20**Published on: **03 Jul '20**Published in: **arXiv - Mathematical Physics**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.