Abstract
We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight λ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for λ ∈ ℕ/2) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.
Article PDF
Similar content being viewed by others
References
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
J. Engelsöy, T. G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
P. Saad, Late time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311 [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G. Penington, S. H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].
E. Witten, Matrix models and deformations of JT gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200582 [arXiv:2006.13414] [INSPIRE].
H. Maxfield and G. J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].
A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
J. Cotler and K. Jensen, AdS3 gravity and random CFT, JHEP 04 (2021) 033 [arXiv:2006.08648] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
T. G. Mertens, G. J. Turiaci and H. L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
F. Ferrari, Gauge theory formulation of hyperbolic gravity, JHEP 03 (2021) 046 [arXiv:2011.02108] [INSPIRE].
L. V. Iliesiu, S. S. Pufu, H. Verlinde and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, JHEP 11 (2019) 091 [arXiv:1905.02726] [INSPIRE].
A. Blommaert, T. G. Mertens and H. Verschelde, The Schwarzian theory — a Wilson line perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
T. G. Mertens and G. J. Turiaci, Defects in Jackiw-Teitelboim quantum gravity, JHEP 08 (2019) 127 [arXiv:1904.05228] [INSPIRE].
G. Sárosi, AdS2 holography and the SYK model, PoS(Modave2017)001 (2018) [arXiv:1711.08482] [INSPIRE].
H. T. Lam, T. G. Mertens, G. J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian quantum mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].
F. M. Haehl and M. Rozali, Fine grained chaos in AdS2 gravity, Phys. Rev. Lett. 120 (2018) 121601 [arXiv:1712.04963] [INSPIRE].
Y.-H. Qi, S.-J. Sin and J. Yoon, Quantum correction to chaos in Schwarzian theory, JHEP 11 (2019) 035 [arXiv:1906.00996] [INSPIRE].
T. G. Mertens, Degenerate operators in JT and Liouville (super)gravity, arXiv:2007.00998 [INSPIRE].
T. G. Mertens and G. J. Turiaci, Liouville quantum gravity — holography, JT and matrices, JHEP 01 (2021) 073 [arXiv:2006.07072] [INSPIRE].
Y. Kimura, JT gravity and the asymptotic Weil-Petersson volume, Phys. Lett. B 811 (2020) 135989 [arXiv:2008.04141] [INSPIRE].
L. J. Mordell, The definite integral \( {\int}_{-\infty}^{\infty}\frac{e^{ax^2+ bx}}{e^{ax}+d} \) da and the analytic theory of numbers and the analytic theory of numbers, Acta Math. 61 (1933) 323.
S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University, Utrecht, The Netherlands (2002) [arXiv:0807.4834] [INSPIRE].
B. Chern and R. C. Rhoades, The Mordell integral, quantum modular forms and mock Jacobi forms, Res. Number Theor. 1 (2015) 1.
F. Olver and W. Rheinbolt, Asymptotics and special functions, Elsevier Science, The Netherlands (2014).
J. L. López and N. M. Temme, Large degree asymptotics of generalized Bernoulli and Euler polynomials, J. Math. Anal. Appl. 363 (2010) 197.
A. Grassi, M. Mariño and S. Zakany, Resumming the string perturbation series, JHEP 05 (2015) 038 [arXiv:1405.4214] [INSPIRE].
A. Blommaert, Dissecting the ensemble in JT gravity, arXiv:2006.13971 [INSPIRE].
Q.-M. Luo and H. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (2005) 290.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2101.06252
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Griguolo, L., Papalini, J. & Seminara, D. On the perturbative expansion of exact bi-local correlators in JT gravity. J. High Energ. Phys. 2021, 140 (2021). https://doi.org/10.1007/JHEP05(2021)140
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)140