Abstract
We study holographic entanglement entropy and holographic complexity in a two-charge, \( \frac{1}{4} \)-BPS family of solutions of type IIB supergravity, controlled by one dimensionless parameter. All the geometries in this family are asymptotically AdS3 ×𝕊3 ×𝕋4 and, varying the parameter that controls them, they interpolate between the global AdS3 × 𝕊3 × 𝕋4 and the massless BTZ3 × 𝕊3 × 𝕋4 geometry. Due to AdS/CFT duality, these geometries are dual to pure CFT heavy states.
We find that there is no emergence of entanglement shadow for all the values of the parameter and we discuss the relation with the massless BTZ result, underlying the relevance of the nature of the dual states.
We also compute the holographic complexity of formation of these geometries, finding a nice monotonic function that interpolates between the pure AdS3 result and the massless BTZ one.
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P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1403.5695] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
M. Alishahiha, Holographic complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) 1 [arXiv:1609.01287] [INSPIRE].
B. Michel and A. Puhm, Corrections in the relative entropy of black hole microstates, JHEP 07 (2018) 179 [arXiv:1801.02615] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
S.G. Avery, Using the D1/D5 CFT to understand black holes, Ph.D. thesis, Ohio State U., Columbus, OH, U.S.A. (2010) [arXiv:1012.0072] [INSPIRE].
V. Balasubramanian, P. Kraus and M. Shigemori, Massless black holes and black rings as effective geometries of the D1-D5 system, Class. Quant. Grav. 22 (2005) 4803 [hep-th/0508110] [INSPIRE].
A. Bombini, A. Galliani, S. Giusto, E. Moscato and R. Russo, Unitary 4-point correlators from classical geometries, Eur. Phys. J. C 78 (2018) 8 [arXiv:1710.06820] [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
B. Freivogel, R. Jefferson, L. Kabir, B. Mosk and I.-S. Yang, Casting shadows on holographic reconstruction, Phys. Rev. D 91 (2015) 086013 [arXiv:1412.5175] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [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].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
S.D. Mathur, A. Saxena and Y.K. Srivastava, Constructing ‘hair’ for the three charge hole, Nucl. Phys. B 680 (2004) 415 [hep-th/0311092] [INSPIRE].
O. Lunin, Adding momentum to D1-D5 system, JHEP 04 (2004) 054 [hep-th/0404006] [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].
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].
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].
K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [arXiv:0706.0216] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [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].
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, L. Martucci, M. Petrini and R. Russo, 6D microstate geometries from 10D structures, Nucl. Phys. B 876 (2013) 509 [arXiv:1306.1745] [INSPIRE].
S. Giusto and R. Russo, Superdescendants of the D1-D5 CFT and their dual 3-charge geometries, JHEP 03 (2014) 007 [arXiv:1311.5536] [INSPIRE].
I. Bena, S. Giusto, M. Shigemori and N.P. Warner, Supersymmetric solutions in six dimensions: a linear structure, JHEP 03 (2012) 084 [arXiv:1110.2781] [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 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].
A. Bombini and S. Giusto, Non-extremal superdescendants of the D1-D5 CFT, JHEP 10 (2017) 023 [arXiv:1706.09761] [INSPIRE].
E. Bakhshaei and A. Bombini, Three-charge superstrata with internal excitations, Class. Quant. Grav. 36 (2019) 055001 [arXiv:1811.00067] [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].
I. Bena, P. Heidmann and D. Turton, AdS2 holography: mind the cap, JHEP 12 (2018) 028 [arXiv:1806.02834] [INSPIRE].
P. Heidmann, Bubbling the NHEK, JHEP 01 (2019) 108 [arXiv:1811.08256] [INSPIRE].
R. Walker, D1-D5-P superstrata in 5 and 6 dimensions: separable wave equations and prepotentials, JHEP 09 (2019) 117 [arXiv:1906.04200] [INSPIRE].
P. Heidmann and N.P. Warner, Superstratum symbiosis, JHEP 09 (2019) 059 [arXiv:1903.07631] [INSPIRE].
S. Giusto and R. Russo, Entanglement entropy and D1-D5 geometries, Phys. Rev. D 90 (2014) 066004 [arXiv:1405.6185] [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].
E. Moscato, Black hole microstates and holography in the D1-D5 CFT, Ph.D. thesis, Queen Mary U., London, U.K., September 2017 [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. 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].
A. Bombini and A. Galliani, AdS3 four-point functions from \( \frac{1}{8} \)-BPS states, JHEP 06 (2019) 044 [arXiv:1904.02656] [INSPIRE].
J. Tian, J. Hou and B. Chen, Holographic correlators on integrable superstrata, Nucl. Phys. B 948 (2019) 114766 [arXiv:1904.04532] [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].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys. 90 (2018) 035007 [arXiv:1801.10352] [INSPIRE].
V. Balasubramanian, A. Lawrence, A. Rolph and S. Ross, Entanglement shadows in LLM geometries, JHEP 11 (2017) 159 [arXiv:1704.03448] [INSPIRE].
S. Carlip, Lectures on (2 + 1) dimensional gravity, J. Korean Phys. Soc. 28 (1995) S447 [gr-qc/9503024] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
A. Reynolds and S.F. Ross, Divergences in holographic complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].
D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the time dependence of holographic complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].
A.P. Reynolds and S.F. Ross, Complexity of the AdS soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
L. Susskind, PiTP lectures on complexity and black holes, arXiv:1808.09941 [INSPIRE].
L. Susskind, Three lectures on complexity and black holes, arXiv:1810.11563 [INSPIRE].
S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [arXiv:1810.05151] [INSPIRE].
V. Balasubramanian, M. DeCross, A. Kar and O. Parrikar, Binding complexity and multiparty entanglement, JHEP 02 (2019) 069 [arXiv:1811.04085] [INSPIRE].
K. Goto, H. Marrochio, R.C. Myers, L. Queimada and B. Yoshida, Holographic complexity equals which action?, JHEP 02 (2019) 160 [arXiv:1901.00014] [INSPIRE].
A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, S.-M. Ruan and J. Simón, First law of holographic complexity, Phys. Rev. Lett. 123 (2019) 081601 [arXiv:1903.04511] [INSPIRE].
S.F. Ross, Complexity and typical microstates, Phys. Rev. D 100 (2019) 066014 [arXiv:1905.06211] [INSPIRE].
M. Cadoni and M. Melis, Holographic entanglement entropy of the BTZ black hole, Found. Phys. 40 (2010) 638 [arXiv:0907.1559] [INSPIRE].
J. de Boer, K. Papadodimas and E. Verlinde, Black hole Berry phase, Phys. Rev. Lett. 103 (2009) 131301 [arXiv:0809.5062] [INSPIRE].
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Bombini, A., Fardelli, G. Holographic entanglement entropy and complexity of microstate geometries. J. High Energ. Phys. 2020, 181 (2020). https://doi.org/10.1007/JHEP06(2020)181
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DOI: https://doi.org/10.1007/JHEP06(2020)181