Gaussian free field and Liouville quantum gravity

TL;DR

The work builds a rigorous mathematical foundation for Liouville quantum gravity by first constructing and analyzing the planar Gaussian free field (GFF) and its continuous Green function, including its distributional nature, circle averages, and conformal invariance. It then develops Gaussian multiplicative chaos (GMC) as a robust framework to define Liouville measures, establishing subcritical existence, convergence, and conformal covariance, and linking these objects to random surfaces and Liouville CFT. Key contributions include a self-contained treatment of the GFF as a random distribution, a detailed construction and analysis of Liouville measures via GMC (including Shamov’s approach), and the demonstration of conformal change-of-variables behavior essential for the quantum geometry interpretation. The results yield a rigorous bridge between probabilistic constructions (GFF, GMC) and the geometric framework of Liouville quantum gravity, enabling a probabilistic treatment of random surfaces and their conformal structure with wide-reaching implications in probability, statistical physics, and mathematical physics.

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.

Figures (34)

• Figure 1: Guide to reading. A solid arrow from Chapter $m$ to Chapter $n$ indicates that $m$ is a preqrequisite for $n$. A dashed arrow indicates a complementary perspective on similar/related topics. The $\simeq$ symbol indicates that Chapter 2 and Chapter 3 are somewhat parallel, with Chapter 2 being focused solely on the construction of the Gaussian Multiplicative Chaos (GMC) measure associated to the (Dirichlet) Gaussian free field, while Chapter 3 gives an exposition of the general theory of GMC.
• Figure 2: A discrete Gaussian free field. Simulation by Oskar-Laurin Koiner.
• Figure 3: The Markovian decomposition of the GFF: here $D$ is a square, and $U\subset D$ a slightly smaller square. The first graph shows $\mathbf{h}_0$, and the second shows $\varphi$. Their sum $\mathbf{h}$ is a GFF in $D$, shown in Figure \ref{['F:dGFF']}. Simulations by Oskar-Laurin Koiner.
• Figure 4: The truncated cones in the construction of the scale invariant auxiliary field. The covariance of the field at $(x,x')$ is obtained by integrating $\mathop{\mathrm{d\!}}\nolimits y \mathop{\mathrm{d\!}}\nolimits t / t^{2}$ in the shaded area.
• Figure 5: A map $\bm$ decorated with loops associated to a set of open edges $\boldsymbol{t}$. a. The map is in blue, with solid open edges and dashed closed edges. b. Open clusters and corresponding open dual clusters are shown in blue and red. c. Every dual vertex is joined to its adjacent primal vertices by a green edge. This results in a refined map $\bar{\bm}$ which is a triangulation. d. The primal and dual open clusters are separated by loops, which are drawn in black and are dashed. Each loop is identified with the set of triangles through which it passes: note that it crosses each triangle in the set exactly once. The oriented root edge of the map is indicated with a blue arrow in subfigures a, b and c. The loop $L_0$ is marked with an arrow in subfigure d, and the arrow indicates the orientation of the loop, parallel to the orientation of the root edge.

Theorems & Definitions (512)

• Definition 1.1: Green function
• Proposition 1.2
• Remark 1.3
• proof
• Remark 1.4
• Definition 1.5: Discrete GFF
• Remark 1.6
• Theorem 1.7
• proof
• Theorem 1.8: Law of the GFF and Dirichlet energy