Around the semi-classical limit of boundary Liouville conformal field theory
Baptiste Cerclé
TL;DR
This work establishes a rigorous semi-classical limit for boundary Liouville CFT on open surfaces, showing that Liouville fields concentrate on a classical minimizer of the Liouville action while second-order fluctuations converge to a massive Gaussian free field with Robin boundary conditions. It develops a robust probabilistic framework using Gaussian Free Fields and derivative Gaussian Multiplicative Chaos to define Liouville correlation functions and the associated stress-energy tensor, proving Ward identities and expressing accessory parameters as derivatives of the classical Liouville action. The results bridge probabilistic Liouville theory with deterministic geometry, enabling CFT-inspired techniques to yield classical higher equations of motion and global Ward identities in the presence of boundaries and conical singularities. The findings illuminate how random geometry described by Liouville CFT collapses to a deterministic geometry in the semi-classical limit and relate accessory parameters and higher equations to the classical action, with potential connections to AdS/CFT through semi-classical symbolics. Overall, the paper provides a rigorous foundation for the deterministic interpretation of Liouville geometry in boundary settings and lays groundwork for further exploration of classical-quantum correspondences in 2D gravity.
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 $γ\in(0,2)$, the semi-classical limit corresponding to taking $γ\to0$. 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.