Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra

We use the SLE_ loop measure to construct a natural representation of the Virasoro algebra of central charge c = c () 1. In particular, we introduce a non-degenerate bilinear Hermitian form (not 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.