We prove smoothness of the correlation functions in probabilistic Liouville Conformal Field Theory. Our result is a step towards proving that the correlation functions satisfy the higher Ward identities and the higher BPZ equations, predicted by the Conformal Bootstrap approach to Conformal Field Theory.
We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere S 2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on S 2. 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 Kupiainen et al. (Commun Math Phys 371:1005-1069, 2019) 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 Oikarinen (Ann Henri Poincaré 20(7):2377-2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.