Abstract
We study the analytic structure of the heavy-heavy-light-light holographic correlators in the supergravity approximation of the AdS3 × S3/CFT2 duality. As an explicit example, we derive the correlator where the heavy operator is a classical microstate of the 5D supersymmetric black hole and its dual geometry interpolates as a function of a continuous parameter between global AdS3 and the extremal BTZ black hole. The simplest perturbation of this interpolating geometry by a light field is described by the Heun equation and we exploit the relation of its connection coefficients to the Liouville CFT to analytically compute the correlator in the two limits, focusing in particular on the black hole regime. In this limit we find that the real poles of the correlator become dense and can be approximated by a cut. We show that, when the charges of the heavy state are in the black hole regime, the discontinuity across the cut has complex poles corresponding to the quasi-normal modes of BTZ. This behaviour is qualitatively similar to what is expected for the large central charge limit of a typical black hole microstate.
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J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [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].
I. Bena et al., Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].
I. Bena et al., Asymptotically-flat supergravity solutions deep inside the black-hole regime, JHEP 02 (2018) 014 [arXiv:1711.10474] [INSPIRE].
M. Shigemori, Superstrata, Gen. Rel. Grav. 52 (2020) 51 [arXiv:2002.01592] [INSPIRE].
J. de Boer, Large N elliptic genus and AdS / CFT correspondence, JHEP 05 (1999) 017 [hep-th/9812240] [INSPIRE].
M. Shigemori, Counting Superstrata, JHEP 10 (2019) 017 [arXiv:1907.03878] [INSPIRE].
D.R. Mayerson and M. Shigemori, Counting D1-D5-P microstates in supergravity, SciPost Phys. 10 (2021) 018 [arXiv:2010.04172] [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [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 et al., Unitary 4-point correlators from classical geometries, Eur. Phys. J. C 78 (2018) 8 [arXiv:1710.06820] [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].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in the proceedings of the 16th International Congress on Mathematical Physics, Prague, Czech Republic, 3–8 August 2009, p. 265–289 [https://doi.org/10.1142/9789814304634_0015] [arXiv:0908.4052] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
G. Aminov, A. Grassi and Y. Hatsuda, Black Hole Quasinormal Modes and Seiberg–Witten Theory, Annales Henri Poincare 23 (2022) 1951 [arXiv:2006.06111] [INSPIRE].
G. Bonelli, C. Iossa, D.P. Lichtig and A. Tanzini, Exact solution of Kerr black hole perturbations via CFT2 and instanton counting: Greybody factor, quasinormal modes, and Love numbers, Phys. Rev. D 105 (2022) 044047 [arXiv:2105.04483] [INSPIRE].
G. Bonelli, C. Iossa, D. Panea Lichtig and A. Tanzini, Irregular Liouville Correlators and Connection Formulae for Heun Functions, Commun. Math. Phys. 397 (2023) 635 [arXiv:2201.04491] [INSPIRE].
M. Bianchi, D. Consoli, A. Grillo and J.F. Morales, QNMs of branes, BHs and fuzzballs from quantum SW geometries, Phys. Lett. B 824 (2022) 136837 [arXiv:2105.04245] [INSPIRE].
M. Bianchi, D. Consoli, A. Grillo and J.F. Morales, More on the SW-QNM correspondence, JHEP 01 (2022) 024 [arXiv:2109.09804] [INSPIRE].
M. Bianchi and G. Di Russo, 2-charge circular fuzz-balls and their perturbations, JHEP 08 (2023) 217 [arXiv:2212.07504] [INSPIRE].
M. Bianchi et al., On the stability and deformability of top stars, arXiv:2305.15105 [INSPIRE].
D. Consoli, F. Fucito, J.F. Morales and R. Poghossian, CFT description of BH’s and ECO’s: QNMs, superradiance, echoes and tidal responses, JHEP 12 (2022) 115 [arXiv:2206.09437] [INSPIRE].
D. Fioravanti and D. Gregori, A new method for exact results on Quasinormal Modes of Black Holes, arXiv:2112.11434 [INSPIRE].
D. Fioravanti and D. Gregori, New Developments in \( \mathcal{N} \) = 2 Supersymmetric Gauge Theories: from Integrability to Black Holes, Acta Phys. Polon. Supp. 16 (2023) 31 [INSPIRE].
M. Dodelson et al., Holographic thermal correlators from supersymmetric instantons, SciPost Phys. 14 (2023) 116 [arXiv:2206.07720] [INSPIRE].
S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, arXiv:2110.05497 [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, arXiv:2112.12156 [INSPIRE].
S. Leutheusser and H. Liu, Subalgebra-subregion duality: emergence of space and time in holography, arXiv:2212.13266 [INSPIRE].
E. Witten, Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?, arXiv:2112.11614 [INSPIRE].
E. Witten, Gravity and the crossed product, JHEP 10 (2022) 008 [arXiv:2112.12828] [INSPIRE].
V. Chandrasekaran, G. Penington and E. Witten, Large N algebras and generalized entropy, arXiv:2209.10454 [https://doi.org/10.1007/JHEP04(2023)009] [INSPIRE].
H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan, Holography for people with no time, SciPost Phys. 14 (2023) 150 [arXiv:2207.00407] [INSPIRE].
H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan, Looking at supersymmetric black holes for a very long time, SciPost Phys. 14 (2023) 128 [arXiv:2207.00408] [INSPIRE].
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS / CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
J.-M. Schlenker and E. Witten, No ensemble averaging below the black hole threshold, JHEP 07 (2022) 143 [arXiv:2202.01372] [INSPIRE].
S. Raju and P. Shrivastava, Critique of the fuzzball program, Phys. Rev. D 99 (2019) 066009 [arXiv:1804.10616] [INSPIRE].
M. Dodelson and A. Zhiboedov, Gravitational orbits, double-twist mirage, and many-body scars, JHEP 12 (2022) 163 [arXiv:2204.09749] [INSPIRE].
M. Kulaxizi, G.S. Ng and A. Parnachev, Black Holes, Heavy States, Phase Shift and Anomalous Dimensions, SciPost Phys. 6 (2019) 065 [arXiv:1812.03120] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Black Holes and Conformal Regge Bootstrap, JHEP 10 (2019) 046 [arXiv:1904.00060] [INSPIRE].
A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from Holography, JHEP 08 (2019) 138 [arXiv:1903.05306] [INSPIRE].
M. Kulaxizi, G.S. Ng and A. Parnachev, Subleading Eikonal, AdS/CFT and Double Stress Tensors, JHEP 10 (2019) 107 [arXiv:1907.00867] [INSPIRE].
Y.-Z. Li, Heavy-light Bootstrap from Lorentzian Inversion Formula, JHEP 07 (2020) 046 [arXiv:1910.06357] [INSPIRE].
Y.-Z. Li and H.-Y. Zhang, More on heavy-light bootstrap up to double-stress-tensor, JHEP 10 (2020) 055 [arXiv:2004.04758] [INSPIRE].
A. Parnachev and K. Sen, Notes on AdS-Schwarzschild eikonal phase, JHEP 03 (2021) 289 [arXiv:2011.06920] [INSPIRE].
R. Karlsson et al., CFT correlators, \( \mathcal{W} \)-algebras and generalized Catalan numbers, JHEP 06 (2022) 162 [arXiv:2111.07924] [INSPIRE].
R. Karlsson, A. Parnachev, V. Prilepina and S. Valach, Thermal stress tensor correlators, OPE and holography, JHEP 09 (2022) 234 [arXiv:2206.05544] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
S. Giusto, M.R.R. Hughes and R. Russo, The Regge limit of AdS3 holographic correlators, JHEP 11 (2020) 018 [arXiv:2007.12118] [INSPIRE].
A. Bombini and A. Galliani, AdS3 four-point functions from \( \frac{1}{8} \)-BPS states, JHEP 06 (2019) 044 [arXiv:1904.02656] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [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].
K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154] [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].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
P. Ronveaux et al., Heun’s Differential Equations, Oxford science publications, Oxford University Press (1995) https://books.google.es/books?id=5p65FD8caCgC.
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
S. Jeong and N. Nekrasov, Opers, surface defects, and Yang-Yang functional, Adv. Theor. Math. Phys. 24 (2020) 1789 [arXiv:1806.08270] [INSPIRE].
M. Piątek and A.R. Pietrykowski, Solving Heun’s equation using conformal blocks, Nucl. Phys. B 938 (2019) 543 [arXiv:1708.06135] [INSPIRE].
K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
L.F. Alday et al., Loop and surface operators in N=2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
K. Ito, S. Kanno and T. Okubo, Quantum periods and prepotential in \( \mathcal{N} \) = 2 SU(2) SQCD, JHEP 08 (2017) 065 [arXiv:1705.09120] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin Systems via β-Deformed Matrix Models, Commun. Math. Phys. 358 (2018) 1041 [arXiv:1104.4016] [INSPIRE].
D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
B. Le Floch, A slow review of the AGT correspondence, J. Phys. A 55 (2022) 353002 [arXiv:2006.14025] [INSPIRE].
M. Matone, Instantons and recursion relations in N=2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].
R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone’s relation in the presence of gravitational couplings, JHEP 04 (2004) 008 [hep-th/0403057] [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].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [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].
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
B. Guo, M.R.R. Hughes, S.D. Mathur and M. Mehta, Contrasting the fuzzball and wormhole paradigms for black holes, Turk. J. Phys. 45 (2021) 281 [arXiv:2111.05295] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
H. Nagoya, Remarks on irregular conformal blocks and Painlevé III and II tau functions, arXiv:1804.04782 [INSPIRE].
R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
G. Bonelli et al., On Painlevé/gauge theory correspondence, Lett. Matth. Phys. 107 (2017) 2359 [arXiv:1612.06235] [INSPIRE].
Acknowledgments
We would like to thank Giulio Bonelli, Daniel Panea Lichtig and Alessandro Tanzini for collaboration at the initial stages of this project and for feedback during its completion. We would like to thank the participants of the workshops “String Theory as a bridge between Gauge Theories and Quantum Gravity” at the University of Turin, “Recent Developments in Quantum Physics of Black Holes” at YITP Kyoto University, and “Black-Hole Microstructure V” at IPhT Sacly for feedback to a preliminary versions of the results presented in this paper. In particular we would like to thank Camillo Imbimbo, Francisco Morales and Andrei Parnchev for discussions.
SG was supported in part by the MIUR-PRIN contract 2017CC72MK003. RR is partially supported by the UK EPSRC grant “CFT and Gravity: Heavy States and Black Holes” EP/W019663/1 and the STFC Consolidated Grants ST/T000686/1.
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Giusto, S., Iossa, C. & Russo, R. The black hole behind the cut. J. High Energ. Phys. 2023, 50 (2023). https://doi.org/10.1007/JHEP10(2023)050
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DOI: https://doi.org/10.1007/JHEP10(2023)050