Liouville quantum gravity -- holography, JT and matrices

TL;DR

This paper explores two-dimensional Liouville gravity and minimal string theory, revealing their connections to quantum deformations of $SL(2, )$, matrix models, and JT gravity, with explicit formulas and comparisons across approaches.

Contribution

It provides explicit formulas for correlators in Liouville gravity with fixed boundary length, and establishes connections with quantum deformations, matrix models, and JT gravity in the large p limit.

Findings

Correlators match between continuum and matrix model approaches in the minimal string case.

Large p limit of minimal string correlators converges to Jackiw-Teitelboim gravity results.

Bulk theory potentially interpretable as a 2D dilaton gravity with sinh potential.

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 $\sinh \Phi$ dilaton potential.