The CFT of SLE loop measures and the Kontsevich--Suhov conjecture

TL;DR

This paper develops a conformal field theory framework for SLE loop measures, constructing Virasoro algebra representations, analyzing their structure, and proving the uniqueness of restriction measures, thus advancing the mathematical understanding of SLE in CFT.

Contribution

It introduces a CFT of SLE loop measures, constructs Virasoro representations, and proves the uniqueness of restriction measures, connecting SLE with conformal field theory rigorously.

Findings

Constructed Virasoro algebra representations on L^2 of SLE loop measure

Proved existence of vanishing singular vectors at arbitrary levels

Established the uniqueness of restriction measures

Abstract

This paper initiates the study of the conformal field theory of the SLE$_\kappa$ loop measure $\nu$ for $\kappa\in(0,4]$, the range where the loop is almost surely simple. First, we construct two commuting representations $(\mathbf{L}_n,\bar{\mathbf{L}}_n)_{n\in\mathbb{Z}}$ of the Virasoro algebra with central charge $c_\mathrm{M}=1-6(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2})^2\leq1$ as (unbounded) first order differential operators on $L^2(\nu)$. Second, we introduce highest-weight representations and characterise their structure: in particular, we prove the existence of vanishing singular vectors at arbitrary levels on the Kac table. Third, we prove an integration by parts formula for the SLE loop measure, and use it to define the Shapovalov form of the representation, a non degenerate (but \emph{not} positive definite) Hermitian form $\mathcal{Q}$ on $L^2(\nu)$ with a remarkably simple geometric expression. The fact that $\mathcal{Q}$ differs from the $L^2(\nu)$-inner product is a manifestation of non-unitarity. Finally, we write down a spectral resolution of $\mathcal{Q}$ using the joint diagonalisation of $\mathbf{L}_0$ and $\bar{\mathbf{L}}_0$. As an application of these results, we provide the first proof of the uniqueness of restriction measures, as conjectured by Kontsevich and Suhov. Our results lay the groundwork for an in-depth study of the CFT of SLE: in forthcoming works, we will define correlation functions on Riemann surfaces, and prove conformal Ward identities, BPZ equations, and conformal bootstrap formulas.