Stress-Energy in Liouville Conformal Field Theory on Compact Riemann Surfaces

Abstract

We derive the conformal Ward identities for the correlation functions of the Stress-Energy tensor in probabilistic Liouville Conformal Field theory on compact Riemann surfaces by varying the correlation functions with respect to the background metric. Conformal symmetry makes it easy to treat variations of the metric that do not change the conformal structure. Variations of the metric that deform the conformal structure have to be treated separately, and this part of the computation relies on regularity and integrability properties of the correlation functions of Liouville theory.