Abstract
AdS2 plays an extremely important role in black-hole physics. We construct several infinite families of supergravity solutions that are asymptotically AdS2 in the UV, and terminate in the IR with a cap that is singular in two dimensions but smooth in ten dimensions. These solutions break conformal invariance, and should correspond to supersymmetric ground states of a holographically dual CFT1. We solve the free massless scalar wave equation on a family of these solutions, finding towers of finite-energy normalizable bound-state excitations. We discuss the intriguing possibility that these excitations correspond to time-dependent excitations of the dual CFT1, which would imply that this CFT1 is dynamical rather than topological, and hence cannot have a conformally invariant ground state.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS 4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
F. Azzurli et al., A universal counting of black hole microstates in AdS 4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
A. Strominger, AdS 2 quantum gravity and string theory, JHEP 01 (1999) 007 [hep-th/9809027] [INSPIRE].
R.K. Gupta and A. Sen, AdS 3 /CF T 2 to AdS 2 /CF T 1, JHEP 04 (2009) 034 [arXiv:0806.0053] [INSPIRE].
V. Balasubramanian, J. de Boer, M.M. Sheikh-Jabbari and J. Simon, What is a chiral 2d CFT? And what does it have to do with extremal black holes?, JHEP 02 (2010) 017 [arXiv:0906.3272] [INSPIRE].
V. Balasubramanian, J. Parsons and S.F. Ross, States of a chiral 2d CFT, Class. Quant. Grav. 28 (2011) 045004 [arXiv:1011.1803] [INSPIRE].
A. Castro and W. Song, Comments on AdS 2 gravity, arXiv:1411.1948 [INSPIRE].
M. Cvetic and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
J.M. Maldacena, J. Michelson and A. Strominger, Anti-de Sitter fragmentation, JHEP 02 (1999) 011 [hep-th/9812073] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 (2015).
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [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].
I. Bena, N. Bobev and N.P. Warner, Bubbles on manifolds with a U(1) isometry, JHEP 08 (2007) 004 [arXiv:0705.3641] [INSPIRE].
O. Lunin, Bubbling geometries for AdS 2 × S 2, JHEP 10 (2015) 167 [arXiv:1507.06670] [INSPIRE].
A. Sen, State operator correspondence and entanglement in AdS 2 /CF T 1, Entropy 13 (2011) 1305 [arXiv:1101.4254] [INSPIRE].
H.J. Boonstra, B. Peeters and K. Skenderis, Brane intersections, Anti-de Sitter space-times and dual superconformal theories, Nucl. Phys. B 533 (1998) 127 [hep-th/9803231] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [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].
I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, JHEP 11 (2006) 042 [hep-th/0608217] [INSPIRE].
I. Bena, C.-W. Wang and N.P. Warner, Plumbing the abyss: black ring microstates, JHEP 07 (2008) 019 [arXiv:0706.3786] [INSPIRE].
I. Bena et al., An infinite-dimensional family of black-hole microstate geometries, JHEP 03 (2011) 022 [Erratum ibid. 04 (2011) 059] [arXiv:1006.3497] [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].
I. Bena et al., Asymptotically-flat supergravity solutions deep inside the black-hole regime, JHEP 02 (2018) 014 [arXiv:1711.10474] [INSPIRE].
G.W. Gibbons and N.P. Warner, Global structure of five-dimensional fuzzballs, Class. Quant. Grav. 31 (2014) 025016 [arXiv:1305.0957] [INSPIRE].
P. de Lange, D.R. Mayerson and B. Vercnocke, Structure of six-dimensional microstate geometries, JHEP 09 (2015) 075 [arXiv:1504.07987] [INSPIRE].
P.A. Haas, Smarr’s formula in 11-dimensional supergravity, Class. Quant. Grav. 35 (2018) 135005 [arXiv:1405.3708] [INSPIRE].
A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric index from black hole entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].
A. Chowdhury, R.S. Garavuso, S. Mondal and A. Sen, Do all BPS black hole microstates carry zero angular momentum?, JHEP 04 (2016) 082 [arXiv:1511.06978] [INSPIRE].
A. Chowdhury, R.S. Garavuso, S. Mondal and A. Sen, BPS state counting in N = 8 supersymmetric string theory for pure D-brane configurations, JHEP 10 (2014) 186 [arXiv:1405.0412] [INSPIRE].
J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
J. Polchinski and M.J. Strassler, The string dual of a confining four-dimensional gauge theory, hep-th/0003136 [INSPIRE].
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 superYang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].
M. Petrini, H. Samtleben, S. Schmidt and K. Skenderis, The 10d uplift of the GPPZ solution, JHEP 07 (2018) 026 [arXiv:1805.01919] [INSPIRE].
N. Bobev, F.F. Gautason, B.E. Niehoff and J. van Muiden, Uplifting GPPZ: a ten-dimensional dual of \( \mathcal{N}={1}^{\ast } \), JHEP 10 (2018) 058 [arXiv:1805.03623] [INSPIRE].
I. Bena et al., Holographic dual of hot Polchinski-Strassler quark-gluon plasma, arXiv:1805.06463 [INSPIRE].
A. Castro, D. Grumiller, F. Larsen and R. McNees, Holographic description of AdS 2 black holes, JHEP 11 (2008) 052 [arXiv:0809.4264] [INSPIRE].
C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, Conformal quantum mechanics as the CFT 1 dual to AdS 2, Phys. Lett. B 701 (2011) 503 [arXiv:1106.0726] [INSPIRE].
R. Jackiw and S.Y. Pi, Conformal blocks for the 4-point function in conformal quantum mechanics, Phys. Rev. D 86 (2012) 045017 [Erratum ibid. D 86 (2012) 089905] [arXiv:1205.0443] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
J.B. Gutowski and H.S. Reall, General supersymmetric AdS 5 black holes, JHEP 04 (2004) 048 [hep-th/0401129] [INSPIRE].
J.P. Gauntlett and J.B. Gutowski, General concentric black rings, Phys. Rev. D 71 (2005) 045002 [hep-th/0408122] [INSPIRE].
I. Bena and N.P. Warner, One ring to rule them all. . . and in the darkness bind them?, Adv. Theor. Math. Phys. 9 (2005) 667 [hep-th/0408106] [INSPIRE].
I. Bena, P. Kraus and N.P. Warner, Black rings in Taub-NUT, Phys. Rev. D 72 (2005) 084019 [hep-th/0504142] [INSPIRE].
I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [INSPIRE].
P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].
F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].
P. Heidmann, Four-center bubbled BPS solutions with a Gibbons-Hawking base, JHEP 10 (2017) 009 [arXiv:1703.10095] [INSPIRE].
I. Bena, P. Heidmann and P.F. Ramirez, A systematic construction of microstate geometries with low angular momentum, JHEP 10 (2017) 217 [arXiv:1709.02812] [INSPIRE].
J. Avila, P.F. Ramirez and A. Ruiperez, One thousand and one bubbles, JHEP 01 (2018) 041 [arXiv:1709.03985] [INSPIRE].
O. Lunin, S.D. Mathur and D. Turton, Adding momentum to supersymmetric geometries, Nucl. Phys. B 868 (2013) 383 [arXiv:1208.1770] [INSPIRE].
S. Giusto and R. Russo, Superdescendants of the D1D5 CFT and their dual 3-charge geometries, JHEP 03 (2014) 007 [arXiv:1311.5536] [INSPIRE].
I. Bena et al., Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].
I. Bena, E. Martinec, D. Turton and N.P. Warner, Momentum fractionation on superstrata, JHEP 05 (2016) 064 [arXiv:1601.05805] [INSPIRE].
I. Bena, E. Martinec, D. Turton and N.P. Warner, M-theory superstrata and the MSW string, JHEP 06 (2017) 137 [arXiv:1703.10171] [INSPIRE].
I. Bena, M. Shigemori and N.P. Warner, Black-hole entropy from supergravity superstrata states, JHEP 10 (2014) 140 [arXiv:1406.4506] [INSPIRE].
J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, Quantizing N = 2 multicenter solutions, JHEP 05 (2009) 002 [arXiv:0807.4556] [INSPIRE].
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.
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].
K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J. de Boer, Six-dimensional supergravity on S 3 × AdS 3 and 2D conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
S.D. Mathur and D. Turton, Microstates at the boundary of AdS, JHEP 05 (2012) 014 [arXiv:1112.6413] [INSPIRE].
S.D. Mathur and D. Turton, Momentum-carrying waves on D1-D5 microstate geometries, Nucl. Phys. B 862 (2012) 764 [arXiv:1202.6421] [INSPIRE].
J.P. Gauntlett et al., All supersymmetric solutions of minimal supergravity in five-dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [INSPIRE].
S. Giusto and R. Russo, Adding new hair to the 3-charge black ring, Class. Quant. Grav. 29 (2012) 085006 [arXiv:1201.2585] [INSPIRE].
O. Vasilakis, Bubbling the newly grown black ring hair, JHEP 05 (2012) 033 [arXiv:1202.1819] [INSPIRE].
I. Bena, S.F. Ross and N.P. Warner, On the oscillation of species, JHEP 09 (2014) 113 [arXiv:1312.3635] [INSPIRE].
I. Bena, N. Bobev and N.P. Warner, Spectral flow and the spectrum of multi-center solutions, Phys. Rev. D 77 (2008) 125025 [arXiv:0803.1203] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
J.M. Bardeen and G.T. Horowitz, The extreme Kerr throat geometry: a vacuum analog of AdS 2 × S 2, Phys. Rev. D 60 (1999) 104030 [hep-th/9905099] [INSPIRE].
A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].
H.K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel. 16 (2013) 8 [arXiv:1306.2517] [INSPIRE].
F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].
S. Giusto, R. Russo and D. Turton, New D1-D5-P geometries from string amplitudes, JHEP 11 (2011) 062 [arXiv:1108.6331] [INSPIRE].
W. Black, R. Russo and D. Turton, The supergravity fields for a D-brane with a travelling wave from string amplitudes, Phys. Lett. B 694 (2010) 246 [arXiv:1007.2856] [INSPIRE].
P.A. Cano and T. Ortin, All the supersymmetric solutions of ungauged \( \mathcal{N}=\left(1,0\right) \) , d = 6 supergravity, arXiv:1804.04945 [INSPIRE].
H. Het Lam and S. Vandoren, BPS solutions of six-dimensional (1, 0) supergravity coupled to tensor multiplets, JHEP 06 (2018) 021 [arXiv:1804.04681] [INSPIRE].
J.B. Gutowski, D. Martelli and H.S. Reall, All supersymmetric solutions of minimal supergravity in six-dimensions, Class. Quant. Grav. 20 (2003) 5049 [hep-th/0306235] [INSPIRE].
S. Giusto, L. Martucci, M. Petrini and R. Russo, 6D microstate geometries from 10D structures, Nucl. Phys. B 876 (2013) 509 [arXiv:1306.1745] [INSPIRE].
O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nucl. Phys. B 610 (2001) 49 [hep-th/0105136] [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].
R. Emparan, D. Mateos and P.K. Townsend, Supergravity supertubes, JHEP 07 (2001) 011 [hep-th/0106012] [INSPIRE].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates, Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, Nucl. Phys. B 710 (2005) 425 [hep-th/0406103] [INSPIRE].
V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [INSPIRE].
S. Giusto, O. Lunin, S.D. Mathur and D. Turton, D1-D5-P microstates at the cap, JHEP 02 (2013) 050 [arXiv:1211.0306] [INSPIRE].
P. Heidmann, in preparation.
V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].
J.M. Maldacena and L. Maoz, Desingularization by rotation, JHEP 12 (2002) 055 [hep-th/0012025] [INSPIRE].
I. Bena, D. Turton, R. Walker and N.P. Warner, Integrability and black-hole microstate geometries, JHEP 11 (2017) 021 [arXiv:1709.01107] [INSPIRE].
A. Bombini et al., Unitary 4-point correlators from classical geometries, Eur. Phys. J. C 78 (2018) 8 [arXiv:1710.06820] [INSPIRE].
A. Tyukov, R. Walker and N.P. Warner, Tidal stresses and energy gaps in microstate geometries, JHEP 02 (2018) 122 [arXiv:1710.09006] [INSPIRE].
S. Raju and P. Shrivastava, A critique of the fuzzball program, arXiv:1804.10616 [INSPIRE].
S. Giusto, E. Moscato and R. Russo, AdS 3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [arXiv:1507.00945] [INSPIRE].
A. Galliani, S. Giusto, E. Moscato and R. Russo, Correlators at large c without information loss, JHEP 09 (2016) 065 [arXiv:1606.01119] [INSPIRE].
A. Galliani, S. Giusto and R. Russo, Holographic 4-point correlators with heavy states, JHEP 10 (2017) 040 [arXiv:1705.09250] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
Y.-Z. Li, S.-L. Li and H. Lü, Exact embeddings of JT gravity in strings and M-theory, Eur. Phys. J. C 78 (2018) 791 [arXiv:1804.09742] [INSPIRE].
K. Skenderis and M. Taylor, Kaluza-Klein holography, JHEP 05 (2006) 057 [hep-th/0603016] [INSPIRE].
E.J. Martinec and S. Massai, String theory of supertubes, JHEP 07 (2018) 163 [arXiv:1705.10844] [INSPIRE].
E.J. Martinec, S. Massai and D. Turton, String dynamics in NS5-F1-P geometries, JHEP 09 (2018) 031 [arXiv:1803.08505] [INSPIRE].
G. Bossard and S. Katmadas, Floating JMaRT, JHEP 04 (2015) 067 [arXiv:1412.5217] [INSPIRE].
I. Bena, G. Bossard, S. Katmadas and D. Turton, Non-BPS multi-bubble microstate geometries, JHEP 02 (2016) 073 [arXiv:1511.03669] [INSPIRE].
I. Bena, G. Bossard, S. Katmadas and D. Turton, Bolting multicenter solutions, JHEP 01 (2017) 127 [arXiv:1611.03500] [INSPIRE].
G. Bossard, S. Katmadas and D. Turton, Two kissing bolts, JHEP 02 (2018) 008 [arXiv:1711.04784] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1806.02834
Dedicated to the memory of Joe Polchinski
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bena, I., Heidmann, P. & Turton, D. AdS2 holography: mind the cap. J. High Energ. Phys. 2018, 28 (2018). https://doi.org/10.1007/JHEP12(2018)028
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2018)028