Les Houches lectures on two-dimensional gravity and holography
Gustavo J. Turiaci
TL;DR
The notes present a cohesive, technical account of exactly computing the gravitational path integral for two-dimensional dilaton gravity (notably Jackiw-Teitelboim gravity) and its supersymmetric extensions, using a BF-theory formulation that localizes on flat connections and reduces the bulk dynamics to boundary Schwarzian modes. They establish and exploit a deep JT/RMT duality: the genus expansion of JT gravity with arbitrary boundaries and topologies matches the correlators of ensembles of random matrices in the double-scaling limit, with the disk, trumpet, and higher-genus geometries providing the building blocks for the matrix-model description. The work then surveys numerous generalizations—deformed dilaton potentials, unorientable spacetimes, fermionic JT gravity, and N=1,2 supergravity—showing how the same torsion and measure techniques yield corresponding matrix ensembles (GOE/GSE/GOE-type, etc.) and how spectral curves encode the dual physics. The conclusions sketch forward paths including end-of-the-world branes, propagating matter, plateau reconstruction, factorization puzzles, and extensions to three-dimensional gravity, highlighting the role of wormholes in quantum gravity and holography. Overall, the paper provides a comprehensive, technically detailed roadmap linking exact 2d gravity path integrals to random matrix theories and their extensions. It also clarifies how topological, boundary, and spin structures shape the dual quantum-mechanical descriptions and their nonperturbative completions. All formulas are presented with precise boundary data and topological couplings, facilitating future analytical or numerical explorations in holographic quantum gravity.
Abstract
Lecture notes prepared for the Les Houches school "Quantum Geometry: Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics" that took place during the summer 2024. We cover the techniques to perform the exact gravitational path integral of two-dimensional dilaton-gravity, and supergravity, over spacetimes with arbitrary topology, with an application to black holes. We discuss the connection with random matrix models and moduli spaces of hyperbolic surfaces briefly, since those concepts were covered in other lectures of the school.