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3d Gravity as a random ensemble

Daniel L. Jafferis, Liza Rozenberg, Gabriel Wong

TL;DR

The paper provides a concrete realization of AdS$_3$ gravity as an ensemble average of 2D CFT data encoded in a GOE matrix-tensor model, reconstructing the 3D topological expansion via a diagrammatic dictionary that maps tensor triple-line graphs to 3-manifolds with Wilson lines. It shows that hyperbolic manifolds reproduce Virasoro TQFT partition functions, while non-hyperbolic cases reveal ensemble-specific eigenvalue statistics, tied to a bulk mapping class-group gauging. The Schwinger-Dyson equations of the matrix-tensor system translate into combinatorial 3-manifold relations, offering a topological recursion-like structure that organizes the $e^{-c}$ gravity expansion. The framework unifies surgery, modular invariance, and boundary/bulk observables, providing a non-perturbative route (via SD equations) toward a complete 3D gravity description consistent with bootstrap constraints. The results point to a rich bridge between random CFT data, VTQFT, and the full topological gravity path integral, with potential extensions to richer symmetries and boundary conditions.

Abstract

We give further evidence that the matrix-tensor model studied in \cite{belin2023} is dual to AdS$_{3}$ gravity including the sum over topologies. This provides a 3D version of the duality between JT gravity and an ensemble of random Hamiltonians, in which the matrix and tensor provide random CFT$_2$ data subject to a potential that incorporates the bootstrap constraints. We show how the Feynman rules of the ensemble produce a sum over all three-manifolds and how surgery is implemented by the matrix integral. The partition functions of the resulting 3d gravity theory agree with Virasoro TQFT (VTQFT) on a fixed, hyperbolic manifold. However, on non-hyperbolic geometries, our 3d gravity theory differs from VTQFT, leading to a difference in the eigenvalue statistics of the associated ensemble. As explained in \cite{belin2023}, the Schwinger-Dyson (SD) equations of the matrix-tensor integral play a crucial role in understanding how gravity emerges in the limit that the ensemble localizes to exact CFT's. We show how the SD equations can be translated into a combinatorial problem about three-manifolds.

3d Gravity as a random ensemble

TL;DR

The paper provides a concrete realization of AdS gravity as an ensemble average of 2D CFT data encoded in a GOE matrix-tensor model, reconstructing the 3D topological expansion via a diagrammatic dictionary that maps tensor triple-line graphs to 3-manifolds with Wilson lines. It shows that hyperbolic manifolds reproduce Virasoro TQFT partition functions, while non-hyperbolic cases reveal ensemble-specific eigenvalue statistics, tied to a bulk mapping class-group gauging. The Schwinger-Dyson equations of the matrix-tensor system translate into combinatorial 3-manifold relations, offering a topological recursion-like structure that organizes the gravity expansion. The framework unifies surgery, modular invariance, and boundary/bulk observables, providing a non-perturbative route (via SD equations) toward a complete 3D gravity description consistent with bootstrap constraints. The results point to a rich bridge between random CFT data, VTQFT, and the full topological gravity path integral, with potential extensions to richer symmetries and boundary conditions.

Abstract

We give further evidence that the matrix-tensor model studied in \cite{belin2023} is dual to AdS gravity including the sum over topologies. This provides a 3D version of the duality between JT gravity and an ensemble of random Hamiltonians, in which the matrix and tensor provide random CFT data subject to a potential that incorporates the bootstrap constraints. We show how the Feynman rules of the ensemble produce a sum over all three-manifolds and how surgery is implemented by the matrix integral. The partition functions of the resulting 3d gravity theory agree with Virasoro TQFT (VTQFT) on a fixed, hyperbolic manifold. However, on non-hyperbolic geometries, our 3d gravity theory differs from VTQFT, leading to a difference in the eigenvalue statistics of the associated ensemble. As explained in \cite{belin2023}, the Schwinger-Dyson (SD) equations of the matrix-tensor integral play a crucial role in understanding how gravity emerges in the limit that the ensemble localizes to exact CFT's. We show how the SD equations can be translated into a combinatorial problem about three-manifolds.
Paper Structure (66 sections, 219 equations, 2 figures)

This paper contains 66 sections, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Summary of our work
  4. A diagrammatics for 3-manifolds
  5. The matrix model and the random statistics of 3d gravity
  6. The Schwinger Dyson equation and sum over 3-manifolds
  7. Outline
  8. Definition of the ensemble
  9. The GOE ensemble and symmetries of $C_{ijk}$'s
  10. The constraint-squared potential
  11. Regularizing the inner product
  12. The tensor model and 4-point crossing
  13. Tensor model propagator and Triple line diagrams
  14. Triple line Feynman diagrams
  15. The matrix model and torus modular invariance
  16. ...and 51 more sections

Figures (2)

  • Figure 1: Gluing two 4-boundary wormholes gives the pillow manifold (without external propagator insertions). This is because the insertion of an Omega loop around an $S^2$ handle effectively fills in the associated non contractible cycle, thereby removing the handle.
  • Figure 2: Explicit demonstration of how to resolve ribbons at junctions into a 2d picture.