Les Houches lectures on two-dimensional gravity and holography

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.

💡 Research Summary

These lecture notes, prepared for the 2024 Les Houches summer school “Quantum Geometry: Mathematical Methods for Gravity, Gauge Theories and Non‑Perturbative Physics,” give a comprehensive, technically detailed account of exact gravitational path‑integral calculations in two‑dimensional dilaton gravity, focusing on Jackiw–Teitelboim (JT) gravity and its supersymmetric and unoriented extensions, and on the dual description in terms of random matrix ensembles.

The authors begin by motivating the study of quantum black holes through the “central dogma” that a black hole of area A behaves as a unitary quantum system with A/4 G_N degrees of freedom. They argue that the most precise laboratory for testing this idea is the Euclidean path integral over metrics and topologies, especially in the AdS₂ context where JT gravity provides a solvable model.

Section 2 introduces the basic JT action
(I_{\rm JT}= -\frac{S_0}{4\pi}\int_M!\sqrt{g},R -\frac12\int_M!\sqrt{g},\Phi(R-\Lambda)+I_{\rm bdy})
with Λ < 0 (AdS₂). The dilaton equation forces constant curvature R = Λ, while the dilaton profile encodes the location of the boundary. By rewriting the theory in first‑order form, the authors exhibit a BF‑type action
(I = -i\int_M !{\rm Tr}, B,F)
with gauge group SL(2,ℝ). The Lagrange multiplier B imposes flatness (F = 0), so the bulk path integral localises on the moduli space of flat SL(2,ℝ) connections. The discussion emphasizes that large diffeomorphisms (the mapping‑class group) and spin‑structure choices are not captured by the gauge symmetry and must be summed over explicitly when one restores the full gravitational path integral.

Boundary conditions are then analysed in detail. Three coordinate patches of AdS₂ are described: the global patch (two asymptotic boundaries, an eternal wormhole), the Poincaré patch (single boundary, vacuum‑like), and the black‑hole (Rindler) patch with a finite temperature β. By fixing the boundary value of the dilaton (φ_b) and the proper length of the boundary circle, the authors derive the Schwarzian effective action
(S_{\rm Sch}= -C\int d\tau,{f,\tau})
which governs the dynamics of the reparameterisation mode f(τ). This action is the low‑energy description of the SYK model and provides the bridge to a quantum mechanical dual.

Section 3 carries out the sum over topologies. The Euler characteristic χ = 2−2g−n controls the weight e^{S_0χ}. For each surface (genus g, n boundaries) the authors compute the one‑loop determinant using Ray‑Singer torsion and the torsion of the flat connection. The result reproduces the genus expansion of a double‑scaled random matrix model. In particular, the disk (χ = 1) gives the leading density of states ρ(E)∼sinh(2π√E) (the Airy‑type spectral curve), while the cylinder (χ = 0) yields the connected two‑boundary correlator that is interpreted as a wormhole contribution. This reproduces the Saad‑Shenker‑Stanford (SSS) JT/RMT duality: the full JT path integral equals the partition function of a Hermitian matrix integral in the double‑scaling limit.

Section 4 generalises the duality to arbitrary dilaton potentials U(Φ). By allowing U(Φ) to be non‑linear, the BF formulation acquires a different gauge algebra (e.g. a Poisson‑sigma model). The authors discuss how spin structures and unoriented surfaces lead to additional sectors corresponding to the orthogonal (β = 1) and symplectic (β = 4) ensembles. They also review recent work on “blunt defects” that modify the spectral curve and generate new non‑perturbative effects.

Section 5 treats supersymmetric extensions. For N = 1 JT supergravity the gauge group becomes OSp(1|2); the boundary theory is a supersymmetric Schwarzian with a fermionic mode ψ(τ). The path integral again localises on flat super‑connections, and the one‑loop torsion reproduces the β = 1 (GOE) matrix ensemble. For N = 2 the gauge group is OSp(2|2) (or equivalently SU(1,1|1)), leading to a richer set of fermionic zero modes and a complex supersymmetric Schwarzian. The authors derive the corresponding matrix model as a β‑ensemble with β = 2 (GUE) or β = 4 (GSE) depending on the choice of spin structure and orientation.

The final section lists open problems: a rigorous non‑perturbative definition of JT gravity (e.g. via complex contour deformations), the role of higher‑genus corrections in the information‑paradox context, extensions to higher‑dimensional near‑extremal black holes, and the interplay between defects, spin structures, and the full classification of admissible matrix ensembles.

Overall, the notes provide a self‑contained, technically thorough roadmap from the classical formulation of 2D dilaton gravity to its exact quantum treatment, the BF‑theoretic localisation, the sum over topologies, and the precise mapping to random matrix theory, together with the supersymmetric and unoriented generalisations that are at the frontier of current research.