Stress-Energy in Liouville Conformal Field Theory

TL;DR

This paper constructs the stress-energy tensor correlation functions in probabilistic Liouville Conformal Field Theory on the sphere, deriving conformal Ward identities and establishing a Virasoro algebra representation.

Contribution

It introduces a new method for defining stress-energy correlations as functional derivatives, enabling control over multiple insertions for representation theory in LCFT.

Findings

Derived conformal Ward identities for LCFT correlation functions.

Constructed a Virasoro algebra representation on the LCFT Hilbert space.

Controlled arbitrary stress-energy tensor insertions using smoothness results.

Abstract

We construct the stress-energy tensor correlation functions in probabilistic Liouville Conformal Field Theory (LCFT) on the two-dimensional sphere by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric. In particular, we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In \cite{ward} the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in \cite{Oik} allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.