Abstract
Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball proposal is to be a viable model for quantum black holes, it should reproduce this expectation. This is considered challenging, because it is natural for the modes on a fuzzball microstate to follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the modes depending not just on the n-quantum number, but also on the J-quantum numbers of the compact dimensions. For a free scalar field ϕ, by imposing a Dirichlet boundary condition ϕ = 0 at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter, motivated by the profile functions of BPS fuzzballs, we consider a generic profile ϕ = ϕ0(θ) instead of ϕ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients of ϕ0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear ramp in our previous work can be understood as arising via an extreme version of level repulsion in such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these aspects from simply putting a cut-off in flat space or AdS (i.e., without a horizon).
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References
S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
S. Das, C. Krishnan, A.P. Kumar and A. Kundu, Synthetic fuzzballs: a linear ramp from black hole normal modes, JHEP 01 (2023) 153 [arXiv:2208.14744] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
V.S. Rychkov, D1-D5 black hole microstate counting from supergravity, JHEP 01 (2006) 063 [hep-th/0512053] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
C. Krishnan and A. Raju, A Note on D1-D5 Entropy and Geometric Quantization, JHEP 06 (2015) 054 [arXiv:1504.04330] [INSPIRE].
I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].
S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena and N.P. Warner, Resolving the Structure of Black Holes: Philosophizing with a Hammer, arXiv:1311.4538 [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [Erratum ibid. 09 (2018) 002] [INSPIRE].
F. Haake, S. Gnutzmann and M. Kus, Quantum Signatures of Chaos, Springer Series in Synergetics, fourth edition, Springer (2018).
S. Das, S.K. Garg, C. Krishnan and A. Kundu, What is the Simplest Linear Ramp?, arXiv:2308.11704 [INSPIRE].
Y.Y. Atas, E. Bogomolny, O. Giraud and G. Roux, Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles, Phys. Rev. Lett. 110 (2013) 084101 [arXiv:1212.5611].
Y. Liu, M.A. Nowak and I. Zahed, Disorder in the Sachdev-Yee-Kitaev Model, Phys. Lett. B 773 (2017) 647 [arXiv:1612.05233] [INSPIRE].
C. Krishnan, S. Sanyal and P.N. Bala Subramanian, Quantum Chaos and Holographic Tensor Models, JHEP 03 (2017) 056 [arXiv:1612.06330] [INSPIRE].
A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor: unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350] [INSPIRE].
C. Krishnan, K.V. Pavan Kumar and S. Sanyal, Random Matrices and Holographic Tensor Models, JHEP 06 (2017) 036 [arXiv:1703.08155] [INSPIRE].
A. Gaikwad and R. Sinha, Spectral Form Factor in Non-Gaussian Random Matrix Theories, Phys. Rev. D 100 (2019) 026017 [arXiv:1706.07439] [INSPIRE].
C. Krishnan, K.V. Pavan Kumar and D. Rosa, Contrasting SYK-like Models, JHEP 01 (2018) 064 [arXiv:1709.06498] [INSPIRE].
R. Bhattacharya, S. Chakrabarti, D.P. Jatkar and A. Kundu, SYK Model, Chaos and Conserved Charge, JHEP 11 (2017) 180 [arXiv:1709.07613] [INSPIRE].
C.V. Johnson, F. Rosso and A. Svesko, Jackiw-Teitelboim supergravity as a double-cut matrix model, Phys. Rev. D 104 (2021) 086019 [arXiv:2102.02227] [INSPIRE].
Y. Chen, Spectral form factor for free large N gauge theory and strings, JHEP 06 (2022) 137 [arXiv:2202.04741] [INSPIRE].
M. B. Marcus and G. Pisier, Annals of Mathematics Studies. Vol. 101: Random Fourier Series with Applications to Harmonic Analysis, Princeton University Press (1981).
B. Bhattacharjee and C. Krishnan, A General Prescription for Semi-Classical Holography, arXiv:1908.04786 [INSPIRE].
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
C. Krishnan and P. Pathak, Normal Modes of the Stretched Horizon: A Bulk Mechanism for Black Hole Microstate Level Spacing, to appear.
S. Das and A. Kundu, to appear.
V. Burman, C. Krishnan and P. Pathak, Normal Modes of de Sitter Space, Hyperfast Scrambling and Type II1 Algebras, to appear.
W.T. Kim, Entropy of (2 + 1)-dimensional de Sitter space in terms of brick wall method, Phys. Rev. D 59 (1999) 047503 [hep-th/9810169] [INSPIRE].
A. Svesko, E. Verheijden, E.P. Verlinde and M.R. Visser, Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity, JHEP 08 (2022) 075 [arXiv:2203.00700] [INSPIRE].
B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons, JHEP 07 (2022) 042 [arXiv:2204.05324] [INSPIRE].
E. Witten, Gravity and the crossed product, JHEP 10 (2022) 008 [arXiv:2112.12828] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
L. Susskind, Entanglement and Chaos in De Sitter Space Holography: An SYK Example, JHAP 1 (2021) 1 [arXiv:2109.14104] [INSPIRE].
C. Krishnan and V. Mohan, Hints of gravitational ergodicity: Berry’s ensemble and the universality of the semi-classical Page curve, JHEP 05 (2021) 126 [arXiv:2102.07703] [INSPIRE].
Acknowledgments
We thank A. Preetham Kumar for crucial contributions in our previous collaboration [5], and Masanori Hanada, Romesh Kaul, Alok Laddha, R. Loganayagam, Ayan Mukhopadhyay, Onkar Parrikar, Ashoke Sen, Kostas Skenderis and Amitabh Virmani for discussions and/or correspondence.
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Das, S., Garg, S.K., Krishnan, C. et al. Fuzzballs and random matrices. J. High Energ. Phys. 2023, 31 (2023). https://doi.org/10.1007/JHEP10(2023)031
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DOI: https://doi.org/10.1007/JHEP10(2023)031