--- title: Liouville Quantum Gravity url: https://www.emergentmind.com/topics/liouville-quantum-gravity type: topic --- # Liouville Quantum Gravity Liouville Quantum Gravity (LQG) is the canonical probabilistic model of random two-dimensional Riemannian geometry, originating in theoretical and mathematical physics as the universal scaling limit of large random planar maps. It extends the formal path-integral of two-dimensional quantum gravity via the exponential of a Gaussian free field (GFF), rigorously implemented using Gaussian multiplicative chaos (GMC) theory. LQG is parametrized by a coupling constant $\gamma\in(0,2)$ (or background charge $Q=2/\gamma+\gamma/2$) and exhibits deep connections to conformal field theory, the KPZ relation, Schramm–Loewner evolution (SLE), and the combinatorics of random maps. The critical case $\gamma=2$ (corresponding to central charge $c=1$), and its extension to supercritical regimes ($\gamma>2$), present further mathematical challenges and distinctive geometry. LQG admits a coordinate-invariant description as an equivalence class of pairs (domain, field), explicit metric and measure structures, a spectrum of canonical surfaces (sphere, disk, cone, wedge), and a coupling with integrable models of random geometry and statistical mechanics. ## 1. Gaussian Free Field Formulation and Conformal Equivalence A base object is the Gaussian free field $h$ on a planar domain $D$ (with Dirichlet or Neumann boundary conditions), a centered Gaussian distribution determined by the Green's function $G_D(z,w)$. As $h$ lacks pointwise interpretation, observables are regularized via circle averages or mollifier convolution at scale $\varepsilon$. LQG is defined not as a single metric space but via equivalence classes: - A **$\gamma$-LQG surface** is an equivalence class of pairs $(D,h)$, where two pairs are equivalent if $h' = h\circ f + Q \log |f'|$ under conformal maps $f:D'\to D$ [$\gamma$ parameter: $Q=2/\gamma+\gamma/2$] [2409.15494, 1502.04343, 1410.7318]. - The construction on arbitrary Riemann surfaces (sphere, disk, torus, annulus) proceeds by mapping to a canonical domain and applying conformal covariance [1504.00625, 1711.06547]. ## 2. Quantum Measures and the KPZ Relation The formal metric tensor $g_{\gamma,h}(z) = e^{\gamma h(z)} (dx^2 + dy^2)$ acquires probabilistic sense via: - **Area measure** (bulk): $\mu_h^\gamma(dz) = \lim_{\varepsilon\to0} \varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)} dz$, with $h_\varepsilon(z)$ the $\varepsilon$-circle average [1908.05573, 2404.16642, 1502.04343]. - **Boundary length**: $\nu_h^\gamma(d\ell) = \lim_{\varepsilon\to0} \varepsilon^{\gamma^2/4} e^{\frac{\gamma}{2}h_\varepsilon(x)} d\ell$ for a smooth boundary arc [1502.04343]. - **Conformal covariance**: measures transform under $(U,h)\to (V,h\circ \phi + Q \log|\phi'|)$ as $\mu_{h\circ \phi + Q\log|\phi'|} = \mu_h \circ \phi$ [2409.15494, 1410.7318, 1908.05573]. The **KPZ relation** encodes the change of fractal dimension between Euclidean and quantum geometries: \[ d_E = (1 - \gamma^2/4)d_Q + (\gamma^2/4) d_Q^2 \qquad \Leftrightarrow \qquad d_Q = \frac{2}{\gamma^2}\left(1 - \sqrt{1 - (\gamma^2/2)d_E }\right) \] [2510.16431, 1410.7318, 2109.01252, 2404.16642, 1903.09111]. ## 3. Metric Geometry and The LQG Distance A canonical random metric is constructed via **Liouville first-passage percolation** (LFPP) and a suitable renormalization: \[ D_h^\varepsilon(z,w) = \inf_{P: z \to w} \int_0^1 e^{\xi h_\varepsilon(P(t))} |P'(t)|dt \] with scaling $a_\varepsilon$ chosen so that $a_\varepsilon^{-1} D_h^\varepsilon$ converges as $\varepsilon\to0$ [2109.01252, 2108.12067]. The limiting metric $D_h$ satisfies: - **Length space**: $D_h$ is a geodesic metric. - **Locality**: Determined by local field. - **Weyl scaling**: $D_{h+f}$ modifies infinitesimal lengths by $e^{\xi f}$. - **Scaling/translation invariance** and **tightness across scales**. For $\gamma<2$, $D_h$ is H\"older continuous with explicit exponents in terms of $\gamma$ and $Q$; for $\gamma=2$, only log-moduli of continuity are achieved. At $\gamma>2$, the metric develops singularities, i.e., points at infinite distance [2109.01252, 2108.12067]. **Critical regime:** For $\gamma=2$ (central charge $c=1$), the metric $D_h$ constructed from the critical LFPP parameter $\xi_*$, any subsequential limit of $a_\varepsilon^{-1} D_h^\varepsilon$ (with $\xi=\xi_*$, $Q=2$) defines a weak LQG metric. The topology is Euclidean, but the modulus of continuity of $D_h(z,w)$ w.r.t. the Euclidean metric is a negative power of log: \[ D_h(z,w) \leq C (\log\frac{1}{|z-w|})^{-\theta},\quad \theta<\xi_*/4;\qquad D_h(z,w) \geq (\log\frac{1}{|z-w|})^{-\theta'},\quad \theta'>3\xi_*/4 \] [2108.12067]. The induced balls are Euclidean open, establishing the precise topological equivalence. ## 4. Canonical Quantum Surfaces and Constructions LQG admits a collection of universal canonical surfaces: - **Quantum sphere**: $(S^2,h)$, the field is a GFF plus constant, conditioned to unit area [1506.03804]. - **Quantum disk**: GFF on disk with boundary conditions, area and boundary length constrained [1502.04343]. - **Quantum cone/wedge**: infinite-volume surfaces, scaling-invariant (cone: field plus $-\alpha \log|z|$; wedge: surfaces with two marked boundary points, field with marked singularities). - **Path-decorated (SLE-decorated) quantum surfaces**: whole-plane LQG coupled to space-filling SLEs, with the peanosphere (mating-of-trees) construction providing a Brownian bridge (or excursion/Lévy) encoding of the quantum boundary lengths and area [2409.15494, 1506.03804]. For $\gamma = \sqrt{8/3}$, the quantum sphere with QLE(8/3, 0) metric is isomorphic— as a measure-metric space— to the Brownian map [1712.01571, 1506.03804]. For general $\gamma<2$, SLE and LQG are entwined via the imaginary geometry framework, and the mating-of-trees (Brownian loop, Bessel, Lévy approaches) underpin the universality and scaling-limit connections to random maps. ## 5. Conformal Field Theory and Correlation Structure Liouville Quantum Gravity is entwined with **Liouville conformal field theory (LCFT)**: - **Action**: $S_L[\phi] = \frac{1}{4\pi} \int \big[ |\nabla \phi|^2 + Q R \phi + 4\pi \mu e^{\gamma \phi} \big]\, d^2z$. - **Vertex operators**: $V_\alpha(z) = e^{\alpha \phi(z)}$, $0<\alpha 2Q$ and $\alpha_i < Q$. - **Ward identities** and BPZ equations: The CFT structure is realized rigorously, with exact expressions and operator product expansions (OPE) (unique three-point structure constants, DOZZ formula) [1410.7318, 1512.01802]. The path-integral approach is made rigorous via GMC, with the partition function and correlators computed as explicit functionals of the field, integrated over the zero mode. Weyl anomaly formulas, modular invariance (torus), and boundary terms (disk, annulus) are established [1504.00625, 1711.06547, 1502.04343]. ## 6. Coupling to Random Planar Maps and Statistical Physics Models LQG arises as scaling limits of large **random planar maps**: - Uniform quadrangulations/triangulations, with appropriately rescaled metric and measure, converge to the sphere/disk quantum surfaces, with area and boundary length measures matching the LQG definitions. - Decorated planar maps (e.g., percolation, Ising, Fortuin-Kasteleyn) correspond to different $\gamma\in(0,2)$ via the relation $c_{\mathrm{matter}} = 25 - 6Q^2$ [2510.16431, 1903.09111]. - SLE curves in the scaling limit of critical lattice models coupled with maps converge to SLE-decorated LQG surfaces [2409.15494, 2510.16431, 1506.03804, 1712.01571]. In the pure gravity case, the metric limit is precisely the Brownian map, and more generally, the KPZ relation links Euclidean and quantum exponents for statistical mechanics models coupled to geometry. For $c_{\mathrm{matter}}>1$, the theory becomes nontrivial but exhibits infinite Hausdorff dimension in the limiting metric [1903.09111]. ## 7. Metric, Spectral, and Geometric Properties - **Hausdorff dimension** of the metric space $(D, D_h)$ is $d_\gamma$, rigorously $d_4=4$ for $\gamma=\sqrt{8/3}$, with upper/lower bounds or conjectures in other cases [2109.01252]. - **Spectral geometry**: The spectrum of the Liouville Laplacian (generator of Liouville Brownian motion) satisfies a Weyl law $N(\lambda)\sim c_\gamma \mu_h(D) \lambda$ ($d_s=2$) [2512.02538, 2307.05407]. The constant $c_\gamma$ is explicit and increases without bound as $\gamma\uparrow 2$. - **Geodesic structure**: For all $\gamma<2$ the metric is geodesic, with unique and merging geodesics at small scales; for $\gamma=2$ the topology remains Euclidean, distances scale as log-powers [2108.12067]. - **KPZ and fractal geometry**: The KPZ relation governs dimensions of sets under quantum and Euclidean metrics/measures. Main open problems include: - Full metric scaling limits for random maps at $0<\gamma<2$ and for $c=1$ ($\gamma=2$) [2108.12067]. - Explicit determination of $d_\gamma$ for general $\gamma$. - Comprehensive treatment of higher-genus surfaces and boundary conditions. - Extension of spectral geometry (eigenvalue statistics, quantum ergodicity) to the random geometry setting [2512.02538, 2307.05407]. --- **References:** - [2510.16431] "Liouville quantum gravity: from random planar maps to conformal field theory" - [1506.03804] "Liouville quantum gravity spheres as matings of finite-diameter trees" - [1712.01571] "Liouville quantum gravity as a metric space and a scaling limit" - [2108.12067] "The critical Liouville quantum gravity metric induces the Euclidean topology" - [2109.01252] "Introduction to the Liouville quantum gravity metric" - [1410.7318] "Liouville Quantum Gravity on the Riemann sphere" - [1502.04343] "Liouville Quantum Gravity on the unit disk" - [1908.05573] "Random surfaces and Liouville quantum gravity" - [1903.09111] "Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach" - [2409.15494] "Reconstructing SLE-decorated Liouville quantum gravity surfaces from random permutons" - [1512.01802] "Local Conformal Structure of Liouville Quantum Gravity" - [2307.05407] "Weyl's law in Liouville quantum gravity" - [2512.02538] "On the spectral geometry of Liouville quantum gravity" - [1711.06547], [1504.00625], [1712.01571], [2411.19810], [1502.04343], [1812.11808], [2006.07072] (for further construction and applications). Source: https://www.emergentmind.com/topics/liouville-quantum-gravity