JT gravity, KdV equations and macroscopic loop operators
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Springer
Received: November 11, 2019 Accepted: January 4, 2020 Published: January 24, 2020
Kazumi Okuyamaa and Kazuhiro Sakaib a
Department of Physics, Shinshu University, 3-1-1 Asahi, Matsumoto 390-8621, Japan b Institute of Physics, Meiji Gakuin University, 1518 Kamikurata-cho, Totsuka-ku, Yokohama 244-8539, Japan
E-mail: [email protected], [email protected] Abstract: We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean AdS2 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. Keywords: 2D Gravity, Matrix Models, Integrable Hierarchies ArXiv ePrint: 1911.01659
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)156
JHEP01(2020)156
JT gravity, KdV equations and macroscopic loop operators
Contents 1 Introduction
1 3 4 6 9 10 15 17
3 Various limits in the low temperature regime 3.1 Low energy expansion of ρ(E) 3.2 ’t Hooft expansion of ZJT 3.3 Low energy expansion of ψ(E) 3.4 WKB expansion of ψ(E) and ρ(E) from topological recursion 3.4.1 WKB expansion of ψ(E) 3.4.2 WKB expansion of ρ(E)
20 20 22 23 25 25 27
4 Numerical analysis of ρ(E) and ψ(E)
28
5 Comment on the spectral form factor
31
6 Conclusions and outlook
32
A Airy case
33
B Partial resummation of the eigenvalue density
36
C String equation for JT gravity
37
D Resolvent and wave functions
39
1
Introduction
The Sachdev-Ye-Kitaev (SYK) model [1–3] and its holographic dual Jackiw-Teitelboim (JT) gravity [4–9] are useful testing ground to study various issues in quantum gravity and holography. This duality is based on the fact that the 1d Schwarzian theory, which arises from the Nambu-Goldstone mode of the spontaneously broken time-reparametrization symmetry of the SYK model, also appears as the boundary mode dynamics of JT gravity on
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JHEP01(2020)156
2 General properties of partition function 2.1 JT gravity as 2d gravity in specific coupling background 2.2 Generalized partition function and KdV constraints 2.3 Lax formalism and master differenti
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