Liouville quantum gravity: from random planar maps to conformal field theory

TL;DR

This work surveys Liouville quantum gravity (LQG) as a rigorous framework for random 2D surfaces, linking the scaling limits of random planar maps to LQG and its coupling with conformal matter via KPZ scaling. It develops the mathematical edifice around Gaussian free fields, Gaussian multiplicative chaos, and the LQG metric, while detailing two complementary connections to conformal field theory: Liouville CFT (LCFT) and its integration with the mating-of-trees framework, yielding exact results for SLE/LQG couplings and boundary observables. The article highlights key results such as the Cardy–Smirnov embedding proving convergence of uniform triangulations to $\sqrt{8/3}$-LQG under discrete conformal embedding, and the DOZZ formula linking LCFT correlation structure to geometric quantities. It closes with an outlook on extending these ideas to minimal string theory, 2D percolation, and Yang–Mills-inspired random surfaces, pointing to a rich interface between probability, geometry, and high-energy physics.

Abstract

Originating in theoretical physics, Liouville quantum gravity (LQG) has been an important topic in probability theory and mathematical physics in the past two decades. In this proceeding, we review two aspects of this topic. The first is that LQG describes the random conformal geometry of the scaling limit of random planar maps. We highlight the convergence of random planar maps under discrete conformal embedding, where couplings between LQG and the Schramm-Loewner evolution (SLE) play a key role. The second aspect is the connection to conformal field theory (CFT). Here we highlight the interplay between Liouville CFT and the SLE/LQG coupling, the CFT description of 2D quantum gravity coupled with conformal matter, and applications to SLE and 2D statistical physics. We conclude with several open questions and future directions.

Figures (5)

• Figure 3.1: Illustration of the definition of a planar map. All planar maps we consider are rooted, meaning that they have a distinguished oriented edge indicated by an arrow. The leftmost and the rightmost planar maps are equivalent viewed as graphs, but viewed as planar maps they are different.
• Figure 3.2: Left: The conformal welding of two quantum disks $S_1$ and $S_2$ sampled independently from $\operatorname{QD}_2$ (conditioned on the event that the two purple curves in the left part of the figure have the same quantum length) is a quantum disk sampled from $\operatorname{QD}_2^W$ with $W=4$ decorated by an independent chordal SLE$_\kappa$$\eta$. Right: The discrete counterpart of the conformal welding result in the left figure for $\kappa=\gamma^2=8/3$. The figure is illustrating that there is a bijection between the objects on the left and right sides of the equality, where on the right side we have a planar map $M$ with disk topology and two marked boundary points, along with a self-avoiding curve $p$ connecting the two marked points.
• Figure 3.3: Left: Illustration of a clockwise space-filling SLE$_\kappa$ loop $\eta$ in $\mathbbm D$ starting and ending at $-i$ for $\kappa\geq 8$. The red curve has quantum length $L_t$ while the blue curve has quantum length $R_t$. Middle: The boundary length process $(L,R)$ of $\eta$ has the law of a 2D Brownian path measure from $(0,1)$ to $(0,0)$ in the first quadrant. Right: The black curves are $L$ and $C-R$ for $C>0$ chosen sufficiently large such that these two curves do not intersect. By identifying points lying on the same (red) horizontal line above $C-R$ or below $L$, or on the same (green) vertical line between the two curves, we obtain the left figure. This procedure is called mating of trees.
• Figure 3.4: Illustration of the Mullin bijection. Top left: A planar map $M$ with a spanning tree $\mathfrak T$. Top middle: The triangulation $T$ with trees $\mathfrak T$ and $\mathfrak T_*$. The green curve indicates the order in which the faces (triangles) of the map were added in the sewing procedure. Top right: The walk $(\mathcal{L},\mathcal{R})$ associated with $(M,\mathfrak T)$ via the Mullin bijection. Bottom: Illustration of the sewing procedure. Before reading the $k$th step of the walk, the part of the map we constructed so far is "below" the line in the left panel. After reading the $k$th step of the walk the map looks like one of the other four panels.
• Figure 3.5: Left: Illustration of the event the $E_a(v)$ appearing in the definition of the Cardy-Smirnov embedding. Middle: An embedding of a planar map into $\Delta$. Right: The percolation cycles of are given by the interfaces between blue and yellow vertices. In the scaling limit these converge to CLE$_6$.

Theorems & Definitions (14)

• Conjecture 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Conjecture 3.5
• Theorem 3.6
• Theorem 3.7
• Definition 4.1
• Definition 4.2
• Definition 4.3