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

TL;DR

This paper derives the conformal Ward identities for Stress--Energy tensor correlations in probabilistic Liouville CFT on compact Riemann surfaces, revealing how these correlations relate to primary field correlations through differential operators.

Contribution

It provides a rigorous derivation of Ward identities in Liouville CFT on compact surfaces, including metric variations affecting conformal structure.

Findings

Correlation functions expressed via differential operators with meromorphic coefficients

Ward identities accommodate metric variations including reparametrizations and conformal deformations

Regularity properties of correlation functions are crucial for handling conformal structure variations

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. The conformal Ward identities show that the correlation functions of the Stress--Energy tensor can be expressed as a differential operator with meromorphic coefficient acting on the correlation functions of the primary fields of Liouville Conformal Field Theory. Variations of the metric come in three different forms: reparametrizations, conformal scalings and deformations of the conformal structure. Conformal symmetry makes it easy to treat variations of the metric that do not deform the conformal structure. Variations 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 Conformal Field Theory.