JT Gravity and the Ensembles of Random Matrix Theory

TL;DR

This work extends the JT gravity–random-matrix correspondence beyond orientable, bosonic setups to include unorientable spacetimes, spin/pin structures, and N=1 supersymmetry. It uncovers a comprehensive mapping between JT (super)gravity and all ten standard random-matrix ensembles (three Dyson and seven Altland-Zirnbauer), with the bulk topological field theory encoding anomalies that align with boundary symmetry classes. The analysis develops and utilizes analytic and combinatorial torsion to define the correct moduli-space measures on unoriented and supersymmetric surfaces, and employs loop equations and Mirzakhani-type recursions to connect genus expansions on both sides. Across multiple cases, explicit disc, trumpet, crosscap, and higher-topology computations reproduce the expected matrix-model correlators, volumes of moduli spaces, and, in several instances, simple recursive relations for supermoduli. The results solidify a unifying framework linking two-dimensional gravity, topological field theories, and the full Altland-Zirnbauer and Dyson symmetry classifications, with clear implications for holography of near-extremal black holes and SYK-like models.

Abstract

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

Figures (22)

• Figure 1: A double trumpet $Y$. At its "center" is a closed geodesic $S$.
• Figure 2: Two trumpets $Y_L$ and $Y_R$. By gluing their "inner boundaries" $S_L$ and $S_R$, one can build a "double trumpet."
• Figure 3: A small neighborhood of $S$ has been cut out of the double trumpet and glued into a two-torus.
• Figure 4: This picture is meant to symbolically convey the idea of building ${\mathbb{RP}}^2$ by closing off a disc with a "crosscap."
• Figure 5: A trumpet that ends on a crosscap; we will call this the crosscap spacetime.