Stress-Energy in Liouville Conformal Field Theory

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.

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