Gaussian free field and Liouville quantum gravity

TL;DR

This paper introduces the mathematical foundations of Liouville quantum gravity, connecting probability and geometry to rigorously analyze random surfaces inspired by string theory's physical models.

Contribution

It provides an accessible introduction to recent rigorous developments in Liouville quantum gravity and its relation to the Gaussian free field.

Findings

Mathematical basis for Liouville quantum gravity established

Verification of key predictions from physics literature

Connections between probability, geometry, and string theory clarified

Abstract

Over fourty years ago, the physicist Polyakov proposed a bold framework for string theory, in which the problem was reduced to the study of certain "random surfaces". He further made the tantalising suggestion that this theory could be explicitly solved. Recent breakthroughs from the last fifteen years have not only given a concrete mathematical basis for this theory but also verified some of its most striking predictions, as well as Polyakov's original vision. This theory, now known in the mathematics literature either as Liouville quantum gravity or Liouville conformal field theory, is based on a remarkable combination of ideas coming from different fields, above all probability and geometry. This book is intended to be an introduction to these developments assuming as few prerequisites as possible.