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JT gravity, KdV equations and macroscopic loop operators

Kazumi Okuyama, Kazuhiro Sakai

TL;DR

This work shows that JT gravity on Euclidean AdS_2 can be formulated as a macroscopic loop observable in a conventional 2d gravity matrix model with a specific infinite-coupling background. By leveraging the Lax pair and KdV constraints, the authors develop a fast, recursive method to compute the genus expansion and a low-temperature 't Hooft expansion, supplementing these with detailed analyses of the eigenvalue density and Baker-Akhiezer function. They obtain high-genus results up to g=46, derive closed-form expressions for leading low-T terms, and reveal an Airy-like all-genus structure governing small-energy behavior, supported by numerical checks. The study also connects to the spectral form factor, illustrating ramp-plateau dynamics in the Airy limit and outlining avenues for nonperturbative completion and generalizations to other JT-related theories. Overall, the work provides a coherent, computationally powerful bridge between JT gravity and old 2d gravity, enabling precise control over both perturbative and certain nonperturbative aspects of the theory.

Abstract

We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean $AdS_2$ background using the matrix model description recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that the partition function of JT gravity is written as the expectation value of a macroscopic loop operator in the old matrix model of 2d gravity in the background where infinitely many couplings are turned on in a specific way. Based on this expression we develop a very efficient method of computing the partition function in the genus expansion as well as in the low temperature expansion by making use of the Korteweg-de Vries constraints obeyed by the partition function. We have computed both these expansions up to very high orders using this method. It turns out that we can take a low temperature limit with the ratio of the temperature and the genus counting parameter held fixed. We find the first few orders of the expansion of the free energy in a closed form in this scaling limit. We also study numerically the behavior of the eigenvalue density and the Baker-Akhiezer function using the results in the scaling limit.

JT gravity, KdV equations and macroscopic loop operators

TL;DR

This work shows that JT gravity on Euclidean AdS_2 can be formulated as a macroscopic loop observable in a conventional 2d gravity matrix model with a specific infinite-coupling background. By leveraging the Lax pair and KdV constraints, the authors develop a fast, recursive method to compute the genus expansion and a low-temperature 't Hooft expansion, supplementing these with detailed analyses of the eigenvalue density and Baker-Akhiezer function. They obtain high-genus results up to g=46, derive closed-form expressions for leading low-T terms, and reveal an Airy-like all-genus structure governing small-energy behavior, supported by numerical checks. The study also connects to the spectral form factor, illustrating ramp-plateau dynamics in the Airy limit and outlining avenues for nonperturbative completion and generalizations to other JT-related theories. Overall, the work provides a coherent, computationally powerful bridge between JT gravity and old 2d gravity, enabling precise control over both perturbative and certain nonperturbative aspects of the theory.

Abstract

We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean background using the matrix model description recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that the partition function of JT gravity is written as the expectation value of a macroscopic loop operator in the old matrix model of 2d gravity in the background where infinitely many couplings are turned on in a specific way. Based on this expression we develop a very efficient method of computing the partition function in the genus expansion as well as in the low temperature expansion by making use of the Korteweg-de Vries constraints obeyed by the partition function. We have computed both these expansions up to very high orders using this method. It turns out that we can take a low temperature limit with the ratio of the temperature and the genus counting parameter held fixed. We find the first few orders of the expansion of the free energy in a closed form in this scaling limit. We also study numerically the behavior of the eigenvalue density and the Baker-Akhiezer function using the results in the scaling limit.

Paper Structure

This paper contains 22 sections, 241 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General properties of partition function
  3. JT gravity as 2d gravity in specific coupling background
  4. Generalized partition function and KdV constraints
  5. Lax formalism and master differential equation
  6. Genus expansion
  7. Genus-zero part of $Z_{\hbox{\scriptsize JT}}$
  8. Low temperature expansion
  9. Various limits in the low temperature regime
  10. Low energy expansion of $\rho(E)$
  11. 't Hooft expansion of $Z_{\hbox{\scriptsize JT}}$
  12. Low energy expansion of $\psi(E)$
  13. WKB expansion of $\psi(E)$ and $\rho(E)$ from topological recursion
  14. WKB expansion of $\psi(E)$
  15. WKB expansion of $\rho(E)$
  16. ...and 7 more sections

Figures (6)

  • Figure 1: Plot of $\rho(E)$ for $\hbar=1/30$. The blue curve represents $\rho(E)$ in the approximation \ref{['eq:rho-approx']} while the orange curve represents the genus-zero eigenvalue density $\rho_0(E)$ in \ref{['eq:rho0']}.
  • Figure 2: Plot of $\rho_{\text{np}}(E)$ for $\hbar=1/30$. Blue dots represent the numerical value of $\rho_{\text{np}}(E)$ obtained from \ref{['eq:rho-approx']} while the red curve is the plot of analytic expression in \ref{['eq:rho-np']}.
  • Figure 3: Plot of $\psi(E)$ for $\hbar=1/30$. Blue dots represent the numerical value of $\psi(E)$ obtained from \ref{['eq:psi-approx']} while the red curve is the plot of analytic expression in \ref{['eq:psi-osc']}.
  • Figure 4: Plot of the real part of ${\cal F}_0(\lambda)$ on the complex $\lambda$-plane.
  • Figure 5: Plot of two-loop correlator in the Airy case for $\hbar=1/10$. The blue curve represents the disconnected part $\langle Z(\beta)^2\rangle_{\text{dis}}$ while the orange curve represents the connected part $\langle Z(\beta)^2\rangle_{\text{conn}}$.
  • ...and 1 more figures