Liouville quantum gravity -- holography, JT and matrices

TL;DR

This work establishes exact disk-level observables in Liouville gravity coupled to minimal matter under fixed-length boundary conditions, revealing a universal gravitational dressing structure for bulk and boundary correlators that mirrors a quantum-deformed SL(2,R) framework. By connecting continuum Liouville gravity to dual matrix models in the (2,p) minimal string and analyzing the JT gravity limit, the authors derive explicit formulas for partition functions and correlators, and formulate a quantum-group perspective with WHittaker/MPS-type vertex functions. They further develop p-deformed Weil-Petersson volumes to describe multi-boundary gluing, and provide evidence that the bulk theory reduces to a 2d dilaton gravity with a sinh Φ potential, offering a coherent bridge between non-critical strings, JT gravity, and matrix-model descriptions. The results illuminate how boundary conditions, operator dressing, and topologies combine to yield JT-like physics while preserving richer Liouville-gravity structure, with potential implications for holography, black-hole microphysics, and the baby-universe Hilbert space. The work also outlines natural extensions to supersymmetric versions and higher-genus topologies, suggesting a robust framework for exploring non-perturbative quantum gravity in two dimensions.

Abstract

We study two-dimensional Liouville gravity and minimal string theory on spaces with fixed length boundaries. We find explicit formulas describing the gravitational dressing of bulk and boundary correlators in the disk. Their structure has a striking resemblance with observables in 2d BF (plus a boundary term), associated to a quantum deformation of $SL(2,\mathbb{R})$, a connection we develop in some detail. For the case of the $(2,p)$ minimal string theory, we compare and match the results from the continuum approach with a matrix model calculation, and verify that in the large $p$ limit the correlators match with Jackiw-Teitelboim gravity. We consider multi-boundary amplitudes that we write in terms of gluing bulk one-point functions using a quantum deformation of the Weil-Petersson volumes and gluing measures. Generating functions for genus zero Weil-Petersson volumes are derived, taking the large $p$ limit. Finally, we present preliminary evidence that the bulk theory can be interpreted as a 2d dilaton gravity model with a $\sinh Φ$ dilaton potential.

Figures (13)

• Figure 1: Genus zero $n$-boundary loop amplitude (here $n=4$).
• Figure 2: Contour deformation from the original one (in green) to a deformed one that wraps the negative real axis (blue line). The segment $(-\kappa,0)$ has no branch cut and the contour can be further deformed to the semi-infinite interval $(-\infty,-\kappa)$.
• Figure 3: FZZT brane segments between $n$ marking operators leads upon transforming to the fixed length basis with length $\ell \equiv \sum_j \ell_j$. In the figure we show an example with $n=3$.
• Figure 4: (Blue) Energy density of states $\rho_0(E_{JT})$ defined in \ref{['defrho0']} with $b=1/2$. (Red) JT limit which focusses on the middle region. (Green) spectral edge limit.
• Figure 5: Matter Coulomb gas two-point function with a vacuum brane $\mathbf{1}$ injected with charge $\beta_M$ to form the state $\beta_M$-brane and then back.