Classifying conformally invariant loop measures

TL;DR

This paper explores the classification of conformally invariant measures on simple loops, connecting concepts from Brownian loops, SLE, and conformal field theory, aiming to deepen understanding of planar statistical mechanics models.

Contribution

It proposes a classification conjecture for conformally invariant loop measures and provides partial algebraic results towards its proof.

Findings

Partial results towards the algebraic classification of loop measures

Connections established between Brownian loop boundaries and conformal invariance

Insights into the relationship between conformal invariance, SLE, and CFT

Abstract

We formulate a classification conjecture for conformally invariant families of measures on simple loops that builds on a conjecture of Kontsevich and Suhov. The main example in this class of objects was constructed by Werner as boundaries of Brownian loops. We present partial results towards the algebraic step of this classification. Solving this conjecture would provide another argument explaining why planar statistical mechanics models with conformally invariant scaling limits naturally occur in a one-parameter family, together with the dynamical characterization of SLE via Schramm's central limit argument, and with the conformal field theory point of view and its central charge parameter.