Abstract
We show that in Jackiw-Teitelboim (JT) gravity, late-time two-point functions can get a leading non-decaying contribution from a spacetime with the topology of a Möbius strip (a disk with one crosscap). There is an interesting interplay between this contribution and the standard “plateau”. The two can add together or cancel, depending on topological weighting factors. We match this behavior to Random Matrix Theory (RMT) and the N mod 8 periodicity of Sachdev-Kitaev-Ye (SYK) results.
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J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].
N. Goheer, M. Kleban and L. Susskind, The trouble with de Sitter space, JHEP 07 (2003) 056 [hep-th/0212209] [INSPIRE].
L. Dyson, M. Kleban and L. Susskind, Disturbing implications of a cosmological constant, JHEP 10 (2002) 011 [hep-th/0208013] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].
P. Saad, Late time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311 [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Clocks and rods in Jackiw-Teitelboim quantum gravity, JHEP 09 (2019) 060 [arXiv:1902.11194] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [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].
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].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian theory — a Wilson line perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [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].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046 [INSPIRE].
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051] [INSPIRE].
A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930 (2018) 45 [arXiv:1712.05073] [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].
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
S. Sachdev and J.-W. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1, talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2, talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
V.V. Belokurov and E.T. Shavgulidze, Exact solution of the Schwarzian theory, Phys. Rev. D 96 (2017) 101701 [arXiv:1705.02405] [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].
A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP 05 (2019) 198 [arXiv:1808.07032] [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].
N. Seiberg and D. Shih, Minimal string theory, Comptes Rendus Physique 6 (2005) 165 [hep-th/0409306] [INSPIRE].
D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24 (2020) 1475 [arXiv:1907.03363] [INSPIRE].
D. Stanford, More quantum noise from wormholes, arXiv:2008.08570 [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
A. Blommaert, Dissecting the ensemble in JT gravity, JHEP 09 (2022) 075 [arXiv:2006.13971] [INSPIRE].
D. Stanford, Z. Yang and S. Yao, Subleading Weingartens, JHEP 02 (2022) 200 [arXiv:2107.10252] [INSPIRE].
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
D. Gomez, A classification of fundamental group elements representing simple closed curves on the punctured Klein bottle, arXiv:1704.02601.
P. Norbury, A new cohomology class on the moduli space of curves, Geom. Topol. 27 (2023) 2695 [arXiv:1712.03662] [INSPIRE].
M. Gendulphe, What’s wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces, arXiv:1706.08798.
Y. Gu, A. Kitaev, S. Sachdev and G. Tarnopolsky, Notes on the complex Sachdev-Ye-Kitaev model, JHEP 02 (2020) 157 [arXiv:1910.14099] [INSPIRE].
V.F. Foit, D. Kabat and G. Lifschytz, Bulk reconstruction for spinor fields in AdS/CFT, JHEP 02 (2020) 129 [arXiv:1912.00952] [INSPIRE].
A. Kitaev, Notes on \( \overset{\sim }{\textrm{SL}} \)(2, R) representations, arXiv:1711.08169 [INSPIRE].
K. Shiozaki, H. Shapourian, K. Gomi and S. Ryu, Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries, Phys. Rev. B 98 (2018) 035151 [arXiv:1710.01886] [INSPIRE].
Acknowledgments
I want to give special thanks to Douglas Stanford for patient guidance, extensive discussions, and inspiring comments throughout this project. I am also grateful to Zhenbin Yang and Shunyu Yao for discussions.
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Yan, C. Crosscap contribution to late-time two-point correlators. J. High Energ. Phys. 2023, 51 (2023). https://doi.org/10.1007/JHEP12(2023)051
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DOI: https://doi.org/10.1007/JHEP12(2023)051