Abstract
Microstrata are the non-extremal analogues of superstrata: they are smooth, non-extremal (non-BPS) solitonic solutions to IIB supergravity whose deep-throat limits approximate black holes. Using perturbation theory and numerical methods, we construct families of solutions using a consistent truncation to three-dimensional supergravity. The most general families presented here involve two continuous parameters, or amplitudes, and four quantized parameters that set the angular momenta and energy levels. Our solutions are asymptotic to the vacuum of the D1-D5 system: AdS3 × S3 × 𝕋4. Using holography, we show that the they are dual to multi-particle states in the D1-D5 CFT involving a large number of mutually non-BPS supergravitons and we determine the anomalous dimensions of these states from the binding energies in supergravity. These binding energies are uniformly negative and depend non-linearly on the amplitudes of the states. In one family of solutions, smoothness restricts some of the fields to lie on a special locus of the parameter space. Using precision holography we show that this special locus can be identified with the multi-particle states constructed via the standard OPE of the single-particle constituents. Our numerical analysis shows that microstrata are robust at large amplitudes and the solutions can be obtained to very high precision.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Bena, E.J. Martinec, S.D. Mathur and N.P. Warner, Snowmass White Paper: Micro- and Macro-Structure of Black Holes, arXiv:2203.04981 [INSPIRE].
I. Bena, E.J. Martinec, S.D. Mathur and N.P. Warner, Fuzzballs and Microstate Geometries: Black-Hole Structure in String Theory, arXiv:2204.13113 [INSPIRE].
P. Heidmann, Non-BPS floating branes and bubbling geometries, JHEP 02 (2022) 162 [arXiv:2112.03279] [INSPIRE].
I. Bah, P. Heidmann and P. Weck, Schwarzschild-like topological solitons, JHEP 08 (2022) 269 [arXiv:2203.12625] [INSPIRE].
I. Bah, D.S. Freed, G.W. Moore, N. Nekrasov, S.S. Razamat and S. Schafer-Nameki, A Panorama Of Physical Mathematics c. 2022, arXiv:2211.04467 [INSPIRE].
I. Bah and P. Heidmann, Geometric Resolution of Schwarzschild Horizon, arXiv:2303.10186 [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].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, in Supersymmetric Mechanics, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
B. Ganchev, A. Houppe and N.P. Warner, Q-balls meet fuzzballs: non-BPS microstate geometries, JHEP 11 (2021) 028 [arXiv:2107.09677] [INSPIRE].
B. Ganchev, S. Giusto, A. Houppe and R. Russo, AdS3 holography for non-BPS geometries, Eur. Phys. J. C 82 (2022) 217 [arXiv:2112.03287] [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].
I. Bena, E. Martinec, D. Turton and N.P. Warner, Momentum Fractionation on Superstrata, JHEP 05 (2016) 064 [arXiv:1601.05805] [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].
E. Bakhshaei and A. Bombini, Three-charge superstrata with internal excitations, Class. Quant. Grav. 36 (2019) 055001 [arXiv:1811.00067] [INSPIRE].
N. Čeplak, R. Russo and M. Shigemori, Supercharging Superstrata, JHEP 03 (2019) 095 [arXiv:1812.08761] [INSPIRE].
P. Heidmann and N.P. Warner, Superstratum Symbiosis, JHEP 09 (2019) 059 [arXiv:1903.07631] [INSPIRE].
P. Heidmann, D.R. Mayerson, R. Walker and N.P. Warner, Holomorphic Waves of Black Hole Microstructure, JHEP 02 (2020) 192 [arXiv:1910.10714] [INSPIRE].
N. Čeplak, S. Hampton and N.P. Warner, Linearizing the BPS equations with vector and tensor multiplets, JHEP 03 (2023) 145 [arXiv:2204.07170] [INSPIRE].
N. Čeplak, Vector Superstrata, JHEP 08 (2023) 047 [arXiv:2212.06947] [INSPIRE].
J. de Boer, Six-dimensional supergravity on S3 × AdS3 and 2-D conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].
D.R. Mayerson, R.A. Walker and N.P. Warner, Microstate Geometries from Gauged Supergravity in Three Dimensions, JHEP 10 (2020) 030 [arXiv:2004.13031] [INSPIRE].
A. Houppe and N.P. Warner, Supersymmetry and superstrata in three dimensions, JHEP 08 (2021) 133 [arXiv:2012.07850] [INSPIRE].
S.R. Coleman, Q-balls, Nucl. Phys. B 262 (1985) 263 [Addendum ibid. 269 (1986) 744] [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, S.F. Ross and N.P. Warner, Coiffured Black Rings, Class. Quant. Grav. 31 (2014) 165015 [arXiv:1405.5217] [INSPIRE].
K. Goldstein and S. Katmadas, Almost BPS black holes, JHEP 05 (2009) 058 [arXiv:0812.4183] [INSPIRE].
I. Bena, G. Dall’Agata, S. Giusto, C. Ruef and N.P. Warner, Non-BPS Black Rings and Black Holes in Taub-NUT, JHEP 06 (2009) 015 [arXiv:0902.4526] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, Multi-Center non-BPS Black Holes: the Solution, JHEP 11 (2009) 032 [arXiv:0908.2121] [INSPIRE].
N. Bobev and C. Ruef, The Nuts and Bolts of Einstein-Maxwell Solutions, JHEP 01 (2010) 124 [arXiv:0912.0010] [INSPIRE].
G. Dall’Agata, S. Giusto and C. Ruef, U-duality and non-BPS solutions, JHEP 02 (2011) 074 [arXiv:1012.4803] [INSPIRE].
O. Vasilakis and N.P. Warner, Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries, JHEP 10 (2011) 006 [arXiv:1104.2641] [INSPIRE].
B. Ganchev, A. Houppe and N.P. Warner, Elliptical and purely NS superstrata, JHEP 09 (2022) 067 [arXiv:2207.04060] [INSPIRE].
B. Ganchev, A. Houppe and N.P. Warner, New superstrata from three-dimensional supergravity, JHEP 04 (2022) 065 [arXiv:2110.02961] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, A (Running) Bolt for New Reasons, JHEP 11 (2009) 089 [arXiv:0909.2559] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, Supergravity Solutions from Floating Branes, JHEP 03 (2010) 047 [arXiv:0910.1860] [INSPIRE].
I. Bena, C. Ruef and N.P. Warner, Imaginary Soaring Branes: A Hidden Feature of Non-Extremal Solutions, JHEP 05 (2012) 143 [arXiv:1105.6255] [INSPIRE].
I. Bena, P. Heidmann, R. Monten and N.P. Warner, Thermal Decay without Information Loss in Horizonless Microstate Geometries, SciPost Phys. 7 (2019) 063 [arXiv:1905.05194] [INSPIRE].
I. Bena, F. Eperon, P. Heidmann and N.P. Warner, The Great Escape: Tunneling out of Microstate Geometries, JHEP 04 (2021) 112 [arXiv:2005.11323] [INSPIRE].
C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, International series in pure and applied mathematics, Springer, New York, U.S.A. (1999), https://doi.org/10.1007/978-1-4757-3069-2.
S. Rawash and D. Turton, Supercharged AdS3 Holography, JHEP 07 (2021) 178 [arXiv:2105.13046] [INSPIRE].
S.G. Avery, Using the D1D5 CFT to Understand Black Holes, Ph.D. thesis, The Ohio State University, Columbus, U.S.A. (2010) [arXiv:1012.0072] [INSPIRE].
S. Hampton, S.D. Mathur and I.G. Zadeh, Lifting of D1-D5-P states, JHEP 01 (2019) 075 [arXiv:1804.10097] [INSPIRE].
K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
S. Giusto, E. Moscato and R. Russo, AdS3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [arXiv:1507.00945] [INSPIRE].
S. Giusto, S. Rawash and D. Turton, Ads3 holography at dimension two, JHEP 07 (2019) 171 [arXiv:1904.12880] [INSPIRE].
N. Ceplak, S. Giusto, M.R.R. Hughes and R. Russo, Holographic correlators with multi-particle states, JHEP 09 (2021) 204 [arXiv:2105.04670] [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N=2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
J.M. Maldacena and A. Strominger, AdS(3) black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, in Astérisque. Vol. S131: Élie Cartan et les mathématiques d’aujourd’hui — Lyon, 25–29 juin 1984, Société mathématique de France (1985), pg. 95, http://www.numdam.org/item/AST_1985_S131_95_0/.
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS / CFT correspondence, hep-th/9908160 [INSPIRE].
G. Arutyunov and S. Frolov, On the correspondence between gravity fields and CFT operators, JHEP 04 (2000) 017 [hep-th/0003038] [INSPIRE].
F. Aprile et al., Single particle operators and their correlators in free \( \mathcal{N} \) = 4 SYM, JHEP 11 (2020) 072 [arXiv:2007.09395] [INSPIRE].
S. Rawash, Black hole microstate geometries and their holographic duals, Ph.D. thesis, Southampton University, Southampton U.K. (2023) [INSPIRE].
O.J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav. 33 (2016) 133001 [arXiv:1510.02804] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
M. Shigemori, Superstrata, Gen. Rel. Grav. 52 (2020) 51 [arXiv:2002.01592] [INSPIRE].
B.D. Chowdhury and S.D. Mathur, Radiation from the non-extremal fuzzball, Class. Quant. Grav. 25 (2008) 135005 [arXiv:0711.4817] [INSPIRE].
B. Chakrabarty, D. Turton and A. Virmani, Holographic description of non-supersymmetric orbifolded D1-D5-P solutions, JHEP 11 (2015) 063 [arXiv:1508.01231] [INSPIRE].
F.C. Eperon, H.S. Reall and J.E. Santos, Instability of supersymmetric microstate geometries, JHEP 10 (2016) 031 [arXiv:1607.06828] [INSPIRE].
B. Chakrabarty, D. Ghosh and A. Virmani, Quasinormal modes of supersymmetric microstate geometries from the D1-D5 CFT, JHEP 10 (2019) 072 [arXiv:1908.01461] [INSPIRE].
B.D. Chowdhury and S.D. Mathur, Pair creation in non-extremal fuzzball geometries, Class. Quant. Grav. 25 (2008) 225021 [arXiv:0806.2309] [INSPIRE].
B.D. Chowdhury and S.D. Mathur, Non-extremal fuzzballs and ergoregion emission, Class. Quant. Grav. 26 (2009) 035006 [arXiv:0810.2951] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Emission from the D1D5 CFT, JHEP 10 (2009) 065 [arXiv:0906.2015] [INSPIRE].
E.J. Martinec and N.P. Warner, The Harder They Fall, the Bigger They Become: Tidal Trapping of Strings by Microstate Geometries, JHEP 04 (2021) 259 [arXiv:2009.07847] [INSPIRE].
N. Ceplak, S. Hampton and Y. Li, Toroidal tidal effects in microstate geometries, JHEP 03 (2022) 021 [arXiv:2106.03841] [INSPIRE].
E.J. Martinec, The holar wind, JHEP 07 (2023) 113 [arXiv:2303.00234] [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, Little Strings, Long Strings, and Fuzzballs, JHEP 11 (2019) 019 [arXiv:1906.11473] [INSPIRE].
T.D. Brennan and E.J. Martinec, Wrapped Fivebranes Redux, JHEP 06 (2021) 011 [arXiv:2012.00790] [INSPIRE].
E.J. Martinec, S. Massai and D. Turton, Stringy Structure at the BPS Bound, JHEP 12 (2020) 135 [arXiv:2005.12344] [INSPIRE].
E.J. Martinec, S. Massai and D. Turton, On the BPS Sector in AdS3/CFT2 Holography, Fortsch. Phys. 71 (2023) 2300015 [arXiv:2211.12476] [INSPIRE].
G.W. Gibbons and N.P. Warner, Global structure of five-dimensional fuzzballs, Class. Quant. Grav. 31 (2014) 025016 [arXiv:1305.0957] [INSPIRE].
I. Bena and N.P. Warner, Resolving the Structure of Black Holes: Philosophizing with a Hammer, arXiv:1311.4538 [INSPIRE].
L.J. Romans, Selfduality for Interacting Fields: Covariant Field Equations for Six-dimensional Chiral Supergravities, Nucl. Phys. B 276 (1986) 71 [INSPIRE].
Acknowledgments
The work of NPW is supported in part by the DOE grant DE-SC0011687. The work of BG, AH and NPW is supported in part by the ERC Grant 787320 — QBH Structure. RR is partially supported by the U.K. EPSRC grant “CFT and Gravity: Heavy States and Black Holes” EP/W019663/1 and the STFC Consolidated Grants ST/T000686/1.
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: 2307.13021
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
Ganchev, B., Giusto, S., Houppe, A. et al. Microstrata. J. High Energ. Phys. 2023, 163 (2023). https://doi.org/10.1007/JHEP10(2023)163
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2023)163