Around the semi-classical limit of boundary Liouville conformal field theory

TL;DR

This paper rigorously analyzes the semi-classical limit of boundary Liouville conformal field theory, demonstrating convergence to a deterministic geometry and connecting probabilistic and classical CFT techniques.

Contribution

It proves the existence of the semi-classical limit in boundary Liouville theory and characterizes it using Gaussian free fields and classical equations of motion.

Findings

Semi-classical limit exists and converges to deterministic geometry

Limit described by massive Gaussian free field with Robin boundary conditions

Classical stress-energy tensor expressed via accessory parameters and Liouville action

Abstract

Liouville conformal field theory describes a random geometry that fluctuates around a deterministic one: the unique solution of the problem of finding, within a given conformal class, a Riemannian metric with prescribed scalar and geodesic curvatures as well as conical singularities and corners. The level of randomness in Liouville theory is measured by the coupling constant $\gamma\in(0,2)$, the semi-classical limit corresponding to taking $\gamma\to0$. Based on the probabilistic definition of Liouville theory, we prove that this semi-classical limit exists and does give rise to this deterministic geometry. At second order this limit is described in terms of a massive Gaussian free field with Robin boundary conditions. This in turn allows to implement CFT-inspired techniques in a deterministic setting: in particular we define the classical stress-energy tensor, show that it can be expressed in terms of accessory parameters (written as regularized derivatives of the Liouville action), and that it gives rise to classical higher equations of motion.