Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra

TL;DR

This paper constructs a representation of the Virasoro algebra with central charge less than or equal to one using SLE loop measures, demonstrating indefinite unitarity through the infinitesimal conformal restriction property.

Contribution

It introduces a new indefinite unitary representation of the Virasoro algebra derived from SLE loop measures, utilizing the conformal restriction property.

Findings

Constructed a non-degenerate Hermitian form from SLE loop measures

Showed the representation is indefinite unitary

Linked the representation to the infinitesimal conformal restriction property

Abstract

We use the SLE$_\kappa$ loop measure to construct a natural representation of the Virasoro algebra of central charge $c = c(\kappa) \le 1$. In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.