A probabilistic approach of Liouville field theory

Rémi Rhodes,Vincent Vargas

doi:10.5802/crphys.43

Abstract

In this article, we present the Liouville field theory, which was introduced in the eighties in physics by Polyakov as a model for fluctuating metrics in 2D quantum gravity, and outline recent mathematical progress in its study. In particular, we explain the probabilistic construction of this theory carried out by David–Kupiainen–Rhodes–Vargas in [] and how this construction connects to the modern and general approach of Conformal Field Theories in physics, called conformal bootstrap and based on representation theory.

Highlights

One of the simplest and at the same time most intriguing 2d Conformal Field Theory (CFT) is Liouville field theory
Speaking and in physics, gravity is described by a Riemannian metric g on a fixed Riemann surface M and quantum gravity can be seen as a way to to sum over the space of Riemannian metrics g on M, which we will call R(M )
The gain in trading (1) for (2) is that, the problem is still infinite-dimensional, the formal definition of the volume form Dφ is linear, which highly increases the possibility of giving a proper construction. In spite of this drastic simplification, this path integral has not been fully understood in physics and has mostly served to heuristically justify inputs in another approach, more algebraic, of Liouville field theory called the conformal bootstrap

Summary

Introduction

One of the simplest and at the same time most intriguing 2d Conformal Field Theory (CFT) is Liouville field theory It first appeared in 1981 in Polyakov’s formulation of bosonic string theory [2] and, since was developed in physics as a model for Euclidean two-dimensional quantum gravity, namely as a way of summing over all possible geometries of a fixed twodimensional manifold. Ten years ago, this model was connected to four dimensional gauge theories via the AGT conjecture [3], experiencing a considerable renewal of interest. We will outline below the ideas that have shaped this rich theory, ranging from original Polyakov’s path integral formulation to the modern approach of conformal bootstrap

Polyakov’s path integral

Discrete random geometries

Probabilistic construction

Conformal bootstrap

Perspectives