Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra

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.