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JT Gravity Limit of Liouville CFT and Matrix Model

Kenta Suzuki, Tadashi Takayanagi

TL;DR

The paper establishes a precise correspondence between Jackiw-Teitelboim gravity on AdS2 and a semiclassical limit of a time-like Liouville–space-like Liouville worldsheet theory, showing exact matches of actions, disk and annulus amplitudes, and the boundary Schwarzian. It identifies a non-perturbative dual description in terms of a time-dependent c=1 matrix model, including a detailed collective-field realization and deformed Fermi surface. The construction is further extended to two-dimensional de Sitter gravity by a simple sign flip of the bulk cosmological constant, with the same matrix-model description governing both AdS2 and dS2. Overall, the work provides a coherent framework linking JT gravity, non-rational 2D string theory, and matrix models, and highlights boundary terms as essential for the correspondence and potential avenues for holography in de Sitter space.

Abstract

In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on two-dimensional anti de-Sitter spaces and a semiclassical limit of $c<1$ two-dimensional string theory. The world-sheet theory of the latter consists of a space-like Liouville CFT coupled to a non-rational CFT defined by a time-like Liouville CFT. We show that their actions, disk partition functions and annulus amplitudes perfectly agree with each other, where the presence of boundary terms plays a crucial role. We also reproduce the boundary Schwarzian theory from the Liouville theory description. Then, we identify a matrix model dual of our two-dimensional string theory with a specific time-dependent background in $c=1$ matrix quantum mechanics. Finally, we also explain the corresponding relation for the two-dimensional de-Sitter JT gravity.

JT Gravity Limit of Liouville CFT and Matrix Model

TL;DR

The paper establishes a precise correspondence between Jackiw-Teitelboim gravity on AdS2 and a semiclassical limit of a time-like Liouville–space-like Liouville worldsheet theory, showing exact matches of actions, disk and annulus amplitudes, and the boundary Schwarzian. It identifies a non-perturbative dual description in terms of a time-dependent c=1 matrix model, including a detailed collective-field realization and deformed Fermi surface. The construction is further extended to two-dimensional de Sitter gravity by a simple sign flip of the bulk cosmological constant, with the same matrix-model description governing both AdS2 and dS2. Overall, the work provides a coherent framework linking JT gravity, non-rational 2D string theory, and matrix models, and highlights boundary terms as essential for the correspondence and potential avenues for holography in de Sitter space.

Abstract

In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on two-dimensional anti de-Sitter spaces and a semiclassical limit of two-dimensional string theory. The world-sheet theory of the latter consists of a space-like Liouville CFT coupled to a non-rational CFT defined by a time-like Liouville CFT. We show that their actions, disk partition functions and annulus amplitudes perfectly agree with each other, where the presence of boundary terms plays a crucial role. We also reproduce the boundary Schwarzian theory from the Liouville theory description. Then, we identify a matrix model dual of our two-dimensional string theory with a specific time-dependent background in matrix quantum mechanics. Finally, we also explain the corresponding relation for the two-dimensional de-Sitter JT gravity.

Paper Structure

This paper contains 22 sections, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. JT Gravity
  3. Disk geometry
  4. Euclidean wormhole geometry
  5. One-loop partition function
  6. Liouville CFT
  7. Liouville action
  8. JT gravity as a semiclassical limit of Liouville action
  9. FZZT brane interpretation
  10. Minisuperspace wavefunctions
  11. Annulus amplitude
  12. Boundary Schwarzian theory
  13. $c=1$ Matrix Model Interpretation
  14. $c=1$ Matrix model and deformation
  15. Collective field description
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Profiles of Fermi surfaces for $t=-4$ (blue), $t=-2$ (red) $t=0$ (green), $t=2$ (orange) and $t=4$ (purple). The Fermi seas extend in the left-hand side of the Fermi surfaces.
  • Figure 2: Structures of Spacetimes of the collective field theory. The left and right picture describe the standard $c=1$ vacuum and our background, described by (\ref{['CFta']}), respectively.
  • Figure 3: Structure of Fermi surfaces paramerized by $\sigma$ and $\tilde{\sigma}$. We also showed momentum functions $p_\pm(x)$.