Uplifting AdS3/CFT2 to flat space holography

  • Adam Ball
  • Elizabeth HimwichEmail author
  • Sruthi A. Narayanan
  • Sabrina Pasterski
  • Andrew Strominger
Open Access
Regular Article - Theoretical Physics


Four-dimensional (4D) flat Minkowski space admits a foliation by hyperbolicslices. Euclidean AdS3 slices fill the past and future lightcones of the origin, while dS3 slices fill the region outside the lightcone. The resulting link between 4D asymptotically flat quantum gravity and AdS3/CFT2 is explored in this paper. The 4D superrotations in the extended BMS4 group are found to act as the familiar conformal transformations on the 3D hyperbolic slices, mapping each slice to itself. The associated 4D superrotation charge is constructed in the covariant phase space formalism. The soft part gives the 2D stress tensor, which acts on the celestial sphere at the boundary of the hyperbolic slices, and is shown to be an uplift to 4D of the familiar 3D holographic AdS3 stress tensor. Finally, we find that 4D quantum gravity contains an unexpected second, conformally soft, dimension (2, 0) mode that is symplectically paired with the celestial stress tensor.


AdS-CFT Correspondence Space-Time Symmetries 


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


  1. [1]
    J. de Boer and S.N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys.B 665 (2003) 545 [hep-th/0303006] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  2. [2]
    M. Campiglia and A. Laddha, Asymptotic symmetries of QED and Weinbergs soft photon theorem, JHEP07 (2015) 115 [arXiv:1505.05346] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, JHEP12 (2015) 094 [arXiv:1509.01406] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP01 (2017) 112 [arXiv:1609.00732] [INSPIRE].
  5. [5]
    M. Gary, S.B. Giddings and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev.D 80 (2009) 085005 [arXiv:0903.4437] [INSPIRE].ADSGoogle Scholar
  6. [6]
    A.L. Fitzpatrick and J. Kaplan, Scattering States in AdS/CFT, arXiv:1104.2597 [INSPIRE].
  7. [7]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
  8. [8]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010) 010 (2010) [Ann. U. Craiova Phys.21 (2011) S11] [arXiv:1102.4632] [INSPIRE].
  9. [9]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
  10. [10]
    G. Barnich and C. Troessaert, Finite BMS transformations, JHEP03 (2016) 167 [arXiv:1601.04090] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    H. Bondi, Gravitational Waves in General Relativity, Nature186 (1960) 535 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    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.ADSzbMATHGoogle Scholar
  13. [13]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev.128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett.119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  17. [17]
    M. Campiglia, Null to time-like infinity Greens functions for asymptotic symmetries in Minkowski spacetime, JHEP11 (2015) 160 [arXiv:1509.01408] [INSPIRE].
  18. [18]
    A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity,” Proc. Roy. Soc. Lond.A 376 (1981) 585.ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    T. Dray and M. Streubel, Angular momentum at null infinity, Class. Quant. Grav.1 (1984) 15 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
  21. [21]
    V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys.208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    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].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  24. [24]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept.323 (2000) 183 [hep-th/9905111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les mathématiques d’aujourd’hui — Lyon, 25-29 juin 1984, no. S131 in Astérisque, pp. 95-116. Société mathématique de France, (1985),
  26. [26]
    C. Fefferman and C.R. Graham, The ambient metric, Ann. Math. Stud.178 (2011) 1 [arXiv:0710.0919] [INSPIRE].Google Scholar
  27. [27]
    G.J. Zuckerman, Action principles and global geometry, Conf. Proc.C 8607214 (1986) 259 [INSPIRE].
  28. [28]
    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in S.W. Hawking and W. Israel eds., Three hundred years of gravitation, (1987), pp. 676-684, [INSPIRE].
  29. [29]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev.D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  31. [31]
    V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev.D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
  32. [32]
    R.M. Wald and A. Zoupas, A general definition ofconserved quantitiesin general relativity and other theories of gravity, Phys. Rev.D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
  33. [33]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys.B 633 (2002) 3 [hep-th/0111246] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    S.G. Avery and B.U.W. Schwab, Noethers second theorem and Ward identities for gauge symmetries, JHEP02 (2016) 031 [arXiv:1510.07038] [INSPIRE].
  35. [35]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinbergs soft graviton theorem, JHEP05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    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].ADSMathSciNetGoogle Scholar
  37. [37]
    L. Donnay, A. Puhm and A. Strominger, Conformally Soft Photons and Gravitons, JHEP01 (2019) 184 [arXiv:1810.05219] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, JHEP12 (2016) 053 [arXiv:1502.06120] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev.D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    D. Christodoulou and S. Klainerman, The Global nonlinear stability of the Minkowski space, Princeton University Press (1993).Google Scholar
  41. [41]
    D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, Adv. Theor. Math. Phys.21 (2017) 1769 [arXiv:1506.02906] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  42. [42]
    A. Nande, M. Pate and A. Strominger, Soft Factorization in QED from 2D Kac-Moody Symmetry, JHEP02 (2018) 079 [arXiv:1705.00608] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett.116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Adam Ball
    • 1
  • Elizabeth Himwich
    • 1
    Email author
  • Sruthi A. Narayanan
    • 1
  • Sabrina Pasterski
    • 1
  • Andrew Strominger
    • 1
  1. 1.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A.

Personalised recommendations