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AdS$_3$ gravity and random CFT

Jordan Cotler, Kristan Jensen

TL;DR

This work computes the Euclidean gravity path integral for AdS3 on spaces with topology T^2×I, isolating a non-saddle, constrained-wormhole contribution that couples boundary tori through a modular sum. The authors show that the resulting two-boundary amplitude matches double-scaled random-matrix statistics with Virasoro symmetry in the low-energy (BTZ-threshold) regime, suggesting that pure AdS3 gravity is dual to an ensemble of CFTs (a 'random CFT') rather than a single theory. They develop a robust phase-space quantization framework, leverage Alekseev–Shatashvili edge-mode dynamics, and perform a careful Zeta-regularized Fourier analysis of the PSL(2,ℤ) sum to reveal the ramp in the spectral form factor and its Virasoro-determined structure. While not proving a complete duality, the results provide strong nonperturbative evidence for an ensemble interpretation of AdS3 gravity and offer a pathway to exploring higher-genus wormholes and potential four-dimensional generalizations.

Abstract

We compute the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. These are Euclidean wormholes, which smoothly interpolate between two asymptotically Euclidean AdS$_3$ regions with torus boundary. From our results we obtain the spectral correlations between BTZ black hole microstates near threshold, as well as extract the spectral form factor at fixed momentum, which has linear growth in time with small fluctuations around it. The low-energy limit of these correlations is precisely that of a double-scaled random matrix ensemble with Virasoro symmetry. Our findings suggest that if pure three-dimensional gravity has a holographic dual, then the dual is an ensemble which generalizes random matrix theory.

AdS$_3$ gravity and random CFT

TL;DR

This work computes the Euclidean gravity path integral for AdS3 on spaces with topology T^2×I, isolating a non-saddle, constrained-wormhole contribution that couples boundary tori through a modular sum. The authors show that the resulting two-boundary amplitude matches double-scaled random-matrix statistics with Virasoro symmetry in the low-energy (BTZ-threshold) regime, suggesting that pure AdS3 gravity is dual to an ensemble of CFTs (a 'random CFT') rather than a single theory. They develop a robust phase-space quantization framework, leverage Alekseev–Shatashvili edge-mode dynamics, and perform a careful Zeta-regularized Fourier analysis of the PSL(2,ℤ) sum to reveal the ramp in the spectral form factor and its Virasoro-determined structure. While not proving a complete duality, the results provide strong nonperturbative evidence for an ensemble interpretation of AdS3 gravity and offer a pathway to exploring higher-genus wormholes and potential four-dimensional generalizations.

Abstract

We compute the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. These are Euclidean wormholes, which smoothly interpolate between two asymptotically Euclidean AdS regions with torus boundary. From our results we obtain the spectral correlations between BTZ black hole microstates near threshold, as well as extract the spectral form factor at fixed momentum, which has linear growth in time with small fluctuations around it. The low-energy limit of these correlations is precisely that of a double-scaled random matrix ensemble with Virasoro symmetry. Our findings suggest that if pure three-dimensional gravity has a holographic dual, then the dual is an ensemble which generalizes random matrix theory.

Paper Structure

This paper contains 30 sections, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Phase space path integrals and "constrain first" quantization
  4. Yang-Mills in Hamiltonian form
  5. Chern-Simons theory
  6. Continuation to imaginary time and Hilbert space trace
  7. Taking stock
  8. First order formulation of AdS$_3$ gravity
  9. Global AdS$_3$ and boundary conditions
  10. Going off-shell
  11. Continuing to imaginary time
  12. What about $SL(2;\mathbb{C})$?
  13. Alekseev-Shatashvili theory
  14. Spectral form factor in random matrix theory
  15. Wormholes
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Chern-Simons theory on $\mathbb{T}^2\times I$ is modular invariant, even when the boundary tori have different complex structures. Here $r$ is the coordinate along the interval, and $(\omega_1(r), \omega_2(r))$ are the lattice vectors corresponding to the torus as a function of $r$.
  • Figure 2: The wormhole in \ref{['E:wormholeRep']}. This is a bottleneck geometry, where the length of the bottleneck is $\pi|b+\bar{b}|$ and spinning fields are twisted by an amount determined by $b-\bar{b}$ after going around it.