Large deviations for random surfaces: the hyperbolic nature of Liouville Field Theory

Abstract

Liouville Field Theory (LFT) is a model of \(2d\) random surfaces involved in \(2d\) string theory or in the description of the fluctuations of metrics in \(2d\) quantum gravity. This is a probabilistic model that consists in weighting the shifted Free Field action with an interaction term involving a cosmological constant \(\mu\) and a background tachyon, which is nothing but a Gaussian multiplicative chaos, formally the exponential of the Free Field times a constant \(\gamma\), called the Liouville conformal factor. We explain how to rigorously construct such a theory on the disk and review some of its properties, like the KPZ formulae. The main input of our work is the study of the semiclassical limit: when sending \(\gamma\) to \(0\) while keeping the quantity \(\Lambda=\mu\gamma^2\) fixed (semiclassical limit regime), we derive exact formulas for the Laplace transform of the Liouville field. Then we prove that this field concentrates on the solution of the Liouville equation with prescribed negative curvature \(8\pi^2\Lambda\): i.e. we prove convergence in probability and characterize the leading fluctuations, which are Gaussian and massive. Though considered as an ansatz in the whole physics literature, it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of LFT. Also, we prove that this description of LFT as an hyperbolic geometry is rather sharp by establishing a large deviation principle with an explicit good rate function. The same analysis is carried out when we further weight the Liouville action with heavy matter operators. This procedure appears when computing the \(n\)-points correlation functions of LFT. We show that the Liouville metric concentrates on metrics with prescribed negative curvature \(8\pi^2\Lambda\) and conical singularities at the places of insertion.