Local Conformal Structure of Liouville Quantum Gravity

Abstract

In 1983 Belavin, Polyakov, and Zamolodchikov (BPZ) formulated the concept of local conformal symmetry in two dimensional quantum field theories. Their ideas had a tremendous impact in physics and mathematics but a rigorous mathematical formulation of their approach has proved elusive. In this work we provide a probabilistic setup to the BPZ approach for the Liouville Conformal Field Theory (LCFT). LCFT has deep connections in physics (string theory, two dimensional gravity) and in mathematics (scaling limits of planar maps, quantum cohomology). We prove the validity of the conformal Ward identities that represent the local conformal symmetry of LCFT and the Belavin–Polyakov–Zamolodchikov differential equations that form the basis for deriving exact formuli for LCFT. We prove several celebrated results on LCFT, in particular an explicit formula for the degenerate 4 point correlation functions leading to a proof of a non trivial functional relation on the 3 point structure constants derived earlier using physical arguments by Teschner. The proofs are based on exact identities for LCFT correlation functions which rely on the underlying Gaussian structure of LCFT combined with estimates from the theory of critical Gaussian Multiplicative Chaos and a careful analysis of singular integrals (Beurling transforms and generalizations). As a by-product, we give bounds on the correlation functions of LCFT when two points collide making rigorous certain predictions from physics on the so-called “operator product expansion” of LCFT. This paper provides themathematical basis for the proof of integrability results for LCFT (the DOZZ conjecture) and the construction of the Virasoro representation theory for LCFT.