JT supergravity and Brezin-Gross-Witten tau-function
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Springer
Received: July Revised: September Accepted: September Published: October
27, 13, 21, 26,
2020 2020 2020 2020
JT supergravity and Brezin-Gross-Witten tau-function
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 thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy. Keywords: 2D Gravity, Integrable Hierarchies, Matrix Models ArXiv ePrint: 2007.09606
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)160
JHEP10(2020)160
Kazumi Okuyamaa and Kazuhiro Sakaib
Contents 1 Introduction
1
2 JT supergravity and Brezin-Gross-Witten τ -function
2 6 6 8 9 11
4 Bessel case 4.1 One-point function 4.2 Bessel kernel and spectral form factor
13 13 16
5 Low temperature expansion
17
6 Large ν regime 6.1 Genus-zero part 6.2 Higher genus free energy and constitutive relation 6.3 Volume of U(ν) 6.4 Correlators at large ν
20 21 24 29 29
7 Conclusions and outlook
32
1
Introduction
Jackiw-Teitelboim (JT) gravity [1, 2] is a simple model of 2d gravity coupled to a scalar field, which serves as a solvable example of the AdS2 /CFT1 correspondence [3–6]. In a recent paper [7], Saad, Shenker, and Stanford showed that the sum over topologies in JT gravity is reproduced from a matrix model, based on the observation of Eynard and Orantin [8] that the recursion relation for the Weil-Petersson volume found by Mirzakhani [9] can be recast as a topological recursion for a certain double-scaled matrix model [10]. This clearly shows that JT gravity on asymptotically AdS2 is holographically dual to an ensemble of one-dimens
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