Abstract
Two-dimensional (2d) field theories invariant under the Bondi-Metzner-Sachs algebra, or 2d BMSFTs in short, are putative holographic duals of Einstein gravity in 3d asymptotically flat spacetimes. When defined on a torus, these field theories come equipped with a modified modular structure. We use the modular covariance of the BMS torus two-point function to develop formulae for different three-point structure constants of the field theory. These structure constants indicate that BMSFTs follow the eigenstate thermalization hypothesis, albeit with some interesting changes to usual 2d CFTs. The singularity structures of the structure constants contain information on perturbations of cosmological horizons in 3d asymptotically flat spacetimes, which we show can also be obtained as a limit of BTZ quasinormal modes.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, AIP Conf. Proc. 484 (1999) 51 [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].
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].
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. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].
S. Banerjee, Null Infinity and Unitary Representation of The Poincare Group, JHEP 01 (2019) 205 [arXiv:1801.10171] [INSPIRE].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [arXiv:2108.04801] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and celestial holography, Phys. Rev. D 107 (2023) 126027 [arXiv:2212.12553] [INSPIRE].
A. Bagchi, S. Banerjee, R. Basu and S. Dutta, Scattering Amplitudes: Celestial and Carrollian, Phys. Rev. Lett. 128 (2022) 241601 [arXiv:2202.08438] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Carrollian Perspective on Celestial Holography, Phys. Rev. Lett. 129 (2022) 071602 [arXiv:2202.04702] [INSPIRE].
A. Bagchi, P. Dhivakar and S. Dutta, AdS Witten diagrams to Carrollian correlators, JHEP 04 (2023) 135 [arXiv:2303.07388] [INSPIRE].
K. Nguyen and P. West, Carrollian conformal fields and flat holography, Universe 9 (2023) 385 [arXiv:2305.02884] [INSPIRE].
A. Saha, Carrollian approach to 1 + 3D flat holography, JHEP 06 (2023) 051 [arXiv:2304.02696] [INSPIRE].
G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
A. Bagchi, S. Detournay and D. Grumiller, Flat-Space Chiral Gravity, Phys. Rev. Lett. 109 (2012) 151301 [arXiv:1208.1658] [INSPIRE].
A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
A. Bagchi and R. Fareghbal, BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].
N.D.S. Gupta, On an analogue of the Galilei group, Nuovo Cim. A 44 (1966) 512.
J.M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaré, Annales de l’I.H.P. Physique théorique 3 (1965) 1.
A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].
G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].
A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 111602 [arXiv:1410.4089] [INSPIRE].
G. Barnich, H.A. Gonzalez, A. Maloney and B. Oblak, One-loop partition function of three-dimensional flat gravity, JHEP 04 (2015) 178 [arXiv:1502.06185] [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].
H. Jiang, W. Song and Q. Wen, Entanglement Entropy in Flat Holography, JHEP 07 (2017) 142 [arXiv:1706.07552] [INSPIRE].
E. Hijano and C. Rabideau, Holographic entanglement and Poincaré blocks in three-dimensional flat space, JHEP 05 (2018) 068 [arXiv:1712.07131] [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].
A. Bagchi and R. Basu, 3D Flat Holography: Entropy and Logarithmic Corrections, JHEP 03 (2014) 020 [arXiv:1312.5748] [INSPIRE].
A. Bagchi, A. Saha and Zodinmawia, BMS Characters and Modular Invariance, JHEP 07 (2019) 138 [arXiv:1902.07066] [INSPIRE].
A. Bagchi, P. Nandi, A. Saha and Zodinmawia, BMS Modular Diaries: Torus one-point function, JHEP 11 (2020) 065 [arXiv:2007.11713] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
M. Srednicki, Thermal fluctuations in quantized chaotic systems, J. Phys. A 29 (1996) L75 [chao-dyn/9511001] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, Rotating Higher Spin Partition Functions and Extended BMS Symmetries, JHEP 04 (2016) 034 [arXiv:1512.03353] [INSPIRE].
A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, BMS Modules in Three Dimensions, Int. J. Mod. Phys. A 31 (2016) 1650068 [arXiv:1603.03812] [INSPIRE].
B. Oblak, BMS Particles in Three Dimensions, Ph.D. thesis, Brussels University, Belgium (2016) [arXiv:1610.08526] [INSPIRE].
A. Bagchi and I. Mandal, On Representations and Correlation Functions of Galilean Conformal Algebras, Phys. Lett. B 675 (2009) 393 [arXiv:0903.4524] [INSPIRE].
L. Cornalba and M.S. Costa, A new cosmological scenario in string theory, Phys. Rev. D 66 (2002) 066001 [hep-th/0203031] [INSPIRE].
L. Cornalba and M.S. Costa, Time dependent orbifolds and string cosmology, Fortsch. Phys. 52 (2004) 145 [hep-th/0310099] [INSPIRE].
B. Jantzen, New proofs for the two Barnes lemmas and an additional lemma, J. Math. Phys. 54 (2013) 012304 [arXiv:1211.2637] [INSPIRE].
M. Becker, Y. Cabrera and N. Su, Finite-temperature three-point function in 2D CFT, JHEP 09 (2014) 157 [arXiv:1407.3415] [INSPIRE].
A. Bagchi et al., Non-Lorentzian chaos and cosmological holography, Phys. Rev. D 104 (2021) L101901 [arXiv:2106.07649] [INSPIRE].
D. Birmingham, Choptuik scaling and quasinormal modes in the AdS / CFT correspondence, Phys. Rev. D 64 (2001) 064024 [hep-th/0101194] [INSPIRE].
D. Birmingham, I. Sachs and S. Sen, Three-dimensional black holes and string theory, Phys. Lett. B 413 (1997) 281 [hep-th/9707188] [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [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, I. Sachs and S.N. Solodukhin, Relaxation in conformal field theory, Hawking-Page transition, and quasinormal normal modes, Phys. Rev. D 67 (2003) 104026 [hep-th/0212308] [INSPIRE].
A. Bagchi, M. Gary and Zodinmawia, Bondi-Metzner-Sachs bootstrap, Phys. Rev. D 96 (2017) 025007 [arXiv:1612.01730] [INSPIRE].
A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS Bootstrap, Class. Quant. Grav. 34 (2017) 174002 [arXiv:1705.05890] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field Theories with Conformal Carrollian Symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
A. Bagchi, R. Basu, A. Mehra and P. Nandi, Field Theories on Null Manifolds, JHEP 02 (2020) 141 [arXiv:1912.09388] [INSPIRE].
J. de Boer et al., Carroll Symmetry, Dark Energy and Inflation, Front. in Phys. 10 (2022) 810405 [arXiv:2110.02319] [INSPIRE].
A. Bagchi et al., Magic fermions: Carroll and flat bands, JHEP 03 (2023) 227 [arXiv:2211.11640] [INSPIRE].
L. Bidussi et al., Fractons, dipole symmetries and curved spacetime, SciPost Phys. 12 (2022) 205 [arXiv:2111.03668] [INSPIRE].
A. Bagchi, K.S. Kolekar and A. Shukla, Carrollian Origins of Bjorken Flow, Phys. Rev. Lett. 130 (2023) 241601 [arXiv:2302.03053] [INSPIRE].
C. Keeler, V.L. Martin and A. Svesko, Connecting quasinormal modes and heat kernels in 1-loop determinants, SciPost Phys. 8 (2020) 017 [arXiv:1811.08433] [INSPIRE].
A. Bagchi, C. Keeler, V.L. Martin and R. Poddar, work in progress.
Acknowledgments
We thank Diptarka Das, Daniel Grumiller, Cynthia Keeler, Victoria Martin, Rahul Poddar for interesting discussions and comments.
During the course of this work, AB was partially supported by a Swarnajayanti fellowship (SB/SJF/2019-20/08) from the Science and Engineering Research Board (SERB) India, the SERB grant (CRG/2020/002035), a visiting professorship at École Polytechnique Paris, a distinguished visiting professorship at NORDITA Stockholm, and a Royal Society of London international exchange grant with the University of Edinburgh. AB also acknowledges the warm hospitality of the Niels Bohr Institute, Copenhagen, the University of Edinburgh, U.K., ULB Brussels and NORDITA, Stockholm, during various stages of this work. SM is supported by grant number 09/092(1039)/2019-EMR-I from Council of Scientific and Industrial Research (CSIR). The research of MR is supported by the Austrian Science Fund (FWF) project P32581. MR also acknowledges the warm hospitality of Harvard University, the Okinawa Institute of Science and Technology (OIST), the Perimeter Institute for Theoretical Physics (PI), and the Yukawa Institute for Theoretical Physics (YITP) during various stages of this work.
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Bagchi, A., Mondal, S., Pal, S. et al. BMS modular covariance and structure constants. J. High Energ. Phys. 2023, 87 (2023). https://doi.org/10.1007/JHEP11(2023)087
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DOI: https://doi.org/10.1007/JHEP11(2023)087