JT gravity, KdV equations and macroscopic loop operators

TL;DR

This paper connects JT gravity's thermal partition function to matrix models, developing efficient computational methods for genus and low-temperature expansions, and analyzing eigenvalue density and Baker-Akhiezer functions.

Contribution

It introduces a novel approach to compute JT gravity partition functions using macroscopic loop operators and KdV constraints, enabling high-order expansions.

Findings

Partition function expressed as expectation value of a macroscopic loop operator

Developed efficient methods for genus and low-temperature expansions

Analyzed eigenvalue density and Baker-Akhiezer function behavior

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.