Abstract
This paper investigates holographic torus correlators of generic operators at conformal infinity and a finite cutoff within AdS3 gravity coupled with a free scalar field. Using a near-boundary analysis and solving the gravitational boundary value problem, we solve Einstein’s equation and calculate mixed correlators for massless and massive coupled scalar fields. The conformal Ward identity on the torus has been reproduced holographically, which can be regarded as a consistency check. Further, recurrence relations for a specific class of higher-point correlators are derived, validating AdS3/CFT2 with non-trivial boundary topology. While the two-point scalar correlator is accurately computed on the thermal AdS3 saddle, the higher-point correlators associated with scalar and stress tensor operators are explored.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
G. Arutyunov and S. Frolov, Three point Green function of the stress energy tensor in the AdS / CFT correspondence, Phys. Rev. D 60 (1999) 026004 [hep-th/9901121] [INSPIRE].
E. D’Hoker et al., Graviton and gauge boson propagators in AdSd+1, Nucl. Phys. B 562 (1999) 330 [hep-th/9902042] [INSPIRE].
S. Raju, Four Point Functions of the Stress Tensor and Conserved Currents in AdS4/CFT3, Phys. Rev. D 85 (2012) 126008 [arXiv:1201.6452] [INSPIRE].
A. Bagchi, D. Grumiller and W. Merbis, Stress tensor correlators in three-dimensional gravity, Phys. Rev. D 93 (2016) 061502 [arXiv:1507.05620] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, Astérisque S131 (1985) 95.
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
K. Skenderis and S.N. Solodukhin, Quantum effective action from the AdS/CFT correspondence, Phys. Lett. B 472 (2000) 316 [hep-th/9910023] [INSPIRE].
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].
C. Fefferman and C.R. Graham, The ambient metric, Ann. Math. Stud. 178 (2011) 1 [arXiv:0710.0919] [INSPIRE].
C.R. Graham and J.M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991) 186 [INSPIRE].
M.T. Anderson, On the structure of conformally compact Einstein metrics, math/0402198.
M.T. Anderson, Geometric aspects of the AdS/CFT correspondence, IRMA Lect. Math. Theor. Phys. 8 (2005) 1 [hep-th/0403087] [INSPIRE].
M.T. Anderson, Einstein metrics with prescribed conformal infinity on 4 manifolds, Geom. Funct. Anal. 18 (2008) 305. [math/0105243] [INSPIRE].
S. He, Y. Li, Y.-Z. Li and Y. Zhang, Holographic torus correlators of stress tensor in AdS3/CFT2, JHEP 06 (2023) 116 [arXiv:2303.13280] [INSPIRE].
S. He and Y. Li, Holographic Euclidean thermal correlator, JHEP 03 (2024) 024 [arXiv:2308.13518] [INSPIRE].
A. Bhatta, S. Chakrabortty, T. Mandal and A. Maurya, Holographic thermal correlators for hyperbolic CFTs, JHEP 11 (2023) 156 [arXiv:2308.14704] [INSPIRE].
G. Georgiou and D. Zoakos, Holographic three-point correlators at finite density and temperature, JHEP 12 (2023) 125 [arXiv:2309.07645] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].
I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T 2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
M. Taylor, \( T\overline{T} \) deformations in general dimensions, Adv. Theor. Math. Phys. 27 (2023) 37 [arXiv:1805.10287] [INSPIRE].
V. Shyam, Finite Cutoff AdS5 Holography and the Generalized Gradient Flow, JHEP 12 (2018) 086 [arXiv:1808.07760] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sarosi, Gravitational path integral from the T 2 deformation, JHEP 09 (2020) 156 [arXiv:2006.01835] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS / CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].
P. Kraus, F. Larsen and R. Siebelink, The gravitational action in asymptotically AdS and flat space-times, Nucl. Phys. B 563 (1999) 259 [hep-th/9906127] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Some Calculable Contributions to Holographic Entanglement Entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].
A. Petkou and K. Skenderis, A nonrenormalization theorem for conformal anomalies, Nucl. Phys. B 561 (1999) 100 [hep-th/9906030] [INSPIRE].
P. Kraus, J. Liu and D. Marolf, Cutoff AdS3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [INSPIRE].
Y. Li and Y. Zhou, Cutoff AdS3 versus \( T\overline{T} \) CFT2 in the large central charge sector: correlators of energy-momentum tensor, JHEP 12 (2020) 168 [arXiv:2005.01693] [INSPIRE].
D. Friedan and S.H. Shenker, The Analytic Geometry of Two-Dimensional Conformal Field Theory, Nucl. Phys. B 281 (1987) 509 [INSPIRE].
T. Eguchi and H. Ooguri, Conformal and Current Algebras on General Riemann Surface, Nucl. Phys. B 282 (1987) 308 [INSPIRE].
J. Polchinski, String theory. Volume 1: An introduction to the bosonic string, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511816079] [INSPIRE].
I. Ichinose and Y. Satoh, Entropies of scalar fields on three-dimensional black holes, Nucl. Phys. B 447 (1995) 340 [hep-th/9412144] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFTd/AdSd+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
V. Cardoso and J.P.S. Lemos, Scalar, electromagnetic and Weyl perturbations of BTZ black holes: Quasinormal modes, Phys. Rev. D 63 (2001) 124015 [gr-qc/0101052] [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. Birmingham, Choptuik scaling and quasinormal modes in the AdS/CFT correspondence, Phys. Rev. D 64 (2001) 064024 [hep-th/0101194] [INSPIRE].
A. Lopez-Ortega and D. Mata-Pacheco, BTZ quasinormal frequencies as poles of Green’s function, arXiv:1806.06547 [INSPIRE].
M. Dodelson et al., Holographic thermal correlators from supersymmetric instantons, SciPost Phys. 14 (2023) 116 [arXiv:2206.07720] [INSPIRE].
Y. Lei, H. Shu, K. Zhang and R.-D. Zhu, Quasinormal modes of C-metric from SCFTs, JHEP 02 (2024) 140 [arXiv:2308.16677] [INSPIRE].
A. Bhatta and T. Mandal, Exact thermal correlators of holographic CFTs, JHEP 02 (2023) 222 [arXiv:2211.02449] [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, US Government printing office (1968).
S. Carlip and C. Teitelboim, Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions, Phys. Rev. D 51 (1995) 622 [gr-qc/9405070] [INSPIRE].
P. Kraus, Lectures on black holes and the AdS3/CFT2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
K. Shiraishi and T. Maki, Quantum fluctuation of stress tensor and black holes in three dimensions, Phys. Rev. D 49 (1994) 5286 [arXiv:1804.07872] [INSPIRE].
G. Lifschytz and M. Ortiz, Scalar field quantization on the (2 + 1)-dimensional black hole background, Phys. Rev. D 49 (1994) 1929 [gr-qc/9310008] [INSPIRE].
E. Keski-Vakkuri, Bulk and boundary dynamics in BTZ black holes, Phys. Rev. D 59 (1999) 104001 [hep-th/9808037] [INSPIRE].
G. Felder and R. Silvotti, Modular Covariance of Minimal Model Correlation Functions, Commun. Math. Phys. 123 (1989) 1 [INSPIRE].
C.-H. Chang, C.-S. Huang and L.-X. Li, W3 Ward identities on a torus, Phys. Lett. B 259 (1991) 267 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
S. He and Y. Sun, Correlation functions of CFTs on a torus with a \( T\overline{T} \) deformation, Phys. Rev. D 102 (2020) 026023 [arXiv:2004.07486] [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].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
M. Banados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
D. Bak, M. Gutperle and S. Hirano, A dilatonic deformation of AdS5 and its field theory dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].
D.Z. Freedman, C. Nunez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].
M. Chiodaroli, J. Estes and Y. Korovin, Holographic two-point functions for Janus interfaces in the D1/D5 CFT, JHEP 04 (2017) 145 [arXiv:1612.08916] [INSPIRE].
D. Bak, M. Gutperle and S. Hirano, Three dimensional Janus and time-dependent black holes, JHEP 02 (2007) 068 [hep-th/0701108] [INSPIRE].
R. Auzzi et al., Volume complexity for Janus AdS3 geometries, JHEP 08 (2021) 045 [arXiv:2105.08729] [INSPIRE].
S.J. Poletti and D.L. Wiltshire, The global properties of static spherically symmetric charged dilaton space-times with a Liouville potential, Phys. Rev. D 50 (1994) 7260 [gr-qc/9407021] [INSPIRE].
K.C.K. Chan and R.B. Mann, Static charged black holes in (2 + 1)-dimensional dilaton gravity, Phys. Rev. D 50 (1994) 6385 [Erratum ibid. 52 (1995) 2600] [gr-qc/9404040] [INSPIRE].
K.C.K. Chan and R.B. Mann, Spinning black holes in (2 + 1)-dimensional string and dilaton gravity, Phys. Lett. B 371 (1996) 199 [gr-qc/9510069] [INSPIRE].
C. Charmousis, Dilaton space-times with a Liouville potential, Class. Quant. Grav. 19 (2002) 83 [hep-th/0107126] [INSPIRE].
C. Charmousis, B. Gouteraux and J. Soda, Einstein-Maxwell-Dilaton theories with a Liouville potential, Phys. Rev. D 80 (2009) 024028 [arXiv:0905.3337] [INSPIRE].
A. Anabalón, H.A. González, A. Neira-Gallegos and J. Oliva, New boundary conditions in Einstein-scalar gravity in three dimensions, JHEP 12 (2023) 149 [arXiv:2307.16027] [INSPIRE].
N.I. Akhiezer, Elements of the theory of elliptic functions, American Mathematical Society (1990) [https://doi.org/10.1090/mmono/079].
Acknowledgments
We want to thank Yi Li, Chen-Te Ma, Juntao Wang, and Long Zhao for their valuable discussions and comments. We are also grateful to all the organizers and participants of the “Quantum Information, Quantum Matter and Quantum Gravity” workshop (YITP-T-23-01) held at YITP, Kyoto University, where a part of this work was done. This work is partly supported by the National Natural Science Foundation of China under Grant No. 12075101 and No. 12235016. S.H. is grateful for financial support from the Fundamental Research Funds for the Central Universities and the Max Planck Partner Group.
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: 2311.09636
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
He, S., Li, YZ. & Zhang, Y. Holographic torus correlators in AdS3 gravity coupled to scalar field. J. High Energ. Phys. 2024, 254 (2024). https://doi.org/10.1007/JHEP05(2024)254
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)254