Zooming in on AdS3/CFT2 near a BPS bound

  • Jelle Hartong
  • Yang Lei
  • Niels Obers
  • Gerben Oling
Open Access
Regular Article - Theoretical Physics


Any (d + 1)-dimensional CFT with a U(1) flavor symmetry, a BPS bound and an exactly marginal coupling admits a decoupling limit in which one zooms in on the spectrum close to the bound. This limit is an Inönü-Wigner contraction of so(2, d+1)⊕u(1) that leads to a relativistic algebra with a scaling generator but no conformal generators. In 2D CFTs, Lorentz boosts are abelian and by adding a second u(1) we find a contraction of two copies of sl(2, ℝ) ⊕ u(1) to two copies of P 2 c , the 2-dimensional centrally extended Poincaré algebra. We show that the bulk is described by a novel non-Lorentzian geometry that we refer to as pseudo-Newton-Cartan geometry. Both the Chern-Simons action on sl(2, ℝ) ⊕ u(1) and the entire phase space of asymptotically AdS3 spacetimes are well-behaved in the corresponding limit if we fix the radial component for the u(1) connection. With this choice, the resulting Newton-Cartan foliation structure is now associated not with time, but with the emerging holographic direction. Since the leaves of this foliation do not mix, the emergence of the holographic direction is much simpler than in AdS3 holography. Furthermore, we show that the asymptotic symmetry algebra of the limit theory consists of a left- and a right-moving warped Virasoro algebra.


AdS-CFT Correspondence Classical Theories of Gravity Conformal Field Theory 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.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].CrossRefzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    M. Kruczenski, Spin chains and string theory, Phys. Rev. Lett. 93 (2004) 161602 [hep-th/0311203] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T. Harmark and M. Orselli, Spin matrix theory: a quantum mechanical model of the AdS/CFT correspondence, JHEP 11 (2014) 134 [arXiv:1409.4417] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].ADSGoogle Scholar
  8. [8]
    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].ADSGoogle Scholar
  10. [10]
    J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J. Hartong, Y. Lei and N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hořava-Lifshitz gravity, Phys. Rev. D 94 (2016) 065027 [arXiv:1604.08054] [INSPIRE].ADSGoogle Scholar
  12. [12]
    G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2 + 1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    E.A. Bergshoeff and J. Rosseel, Three-dimensional extended Bargmann supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D.M. Hofman and B. Rollier, Warped conformal field theory as lower spin gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    D.M. Hofman and A. Strominger, Chiral scale and conformal invariance in 2D quantum field theory, Phys. Rev. Lett. 107 (2011) 161601 [arXiv:1107.2917] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    S. Detournay, T. Hartman and D.M. Hofman, Warped conformal field theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [INSPIRE].ADSGoogle Scholar
  17. [17]
    S. El-Showk and M. Guica, Kerr/CFT, dipole theories and nonrelativistic CFTs, JHEP 12 (2012) 009 [arXiv:1108.6091] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    W. Song and A. Strominger, Warped AdS 3 /dipole-CFT duality, JHEP 05 (2012) 120 [arXiv:1109.0544] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    W. Song and J. Xu, Correlation functions of warped CFT, JHEP 04 (2018) 067 [arXiv:1706.07621] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Jensen, Locality and anomalies in warped conformal field theory, JHEP 12 (2017) 111 [arXiv:1710.11626] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].ADSzbMATHGoogle Scholar
  22. [22]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Hartong, E. Kiritsis and N.A. Obers, Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].CrossRefzbMATHGoogle Scholar
  25. [25]
    U. Gürsoy, Improved holographic QCD and the quark-gluon plasma, Acta Phys. Polon. B 47 (2016) 2509 [arXiv:1612.00899] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers, Homogeneous nonrelativistic geometries as coset spaces, arXiv:1712.03980 [INSPIRE].
  27. [27]
    K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    G. Papageorgiou and B.J. Schroers, Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra, JHEP 11 (2010) 020 [arXiv:1008.0279] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian gravity and the Bargmann algebra, Class. Quant. Grav. 28 (2011) 105011 [arXiv:1011.1145] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
  33. [33]
    K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
  34. [34]
    J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].ADSGoogle Scholar
  35. [35]
    J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].ADSGoogle Scholar
  37. [37]
    T. Griffin, P. Hořava and C.M. Melby-Thompson, Lifshitz gravity for Lifshitz holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    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].ADSCrossRefzbMATHGoogle Scholar
  40. [40]
    M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, arXiv:1710.08415 [INSPIRE].
  41. [41]
    A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  43. [43]
    E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
  44. [44]
    A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    A. Maloney, Geometric microstates for the three dimensional black hole?, arXiv:1508.04079 [INSPIRE].
  46. [46]
    C.R. Nappi and E. Witten, A WZW model based on a nonsemisimple group, Phys. Rev. Lett. 71 (1993) 3751 [hep-th/9310112] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    A. Bagchi, Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M. Gary, D. Grumiller, M. Riegler and J. Rosseel, Flat space (higher spin) gravity with chemical potentials, JHEP 01 (2015) 152 [arXiv:1411.3728] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    R. Basu, S. Detournay and M. Riegler, Spectral flow in 3D flat spacetimes, JHEP 12 (2017) 134 [arXiv:1706.07438] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  51. [51]
    J. Figueroa-O’Farrill, Classification of kinematical Lie algebras, arXiv:1711.05676 [INSPIRE].
  52. [52]
    J.M. Figueroa-O’Farrill, Kinematical Lie algebras via deformation theory, arXiv:1711.06111 [INSPIRE].
  53. [53]
    J.M. Figueroa-O’Farrill, Higher-dimensional kinematical Lie algebras via deformation theory, arXiv:1711.07363 [INSPIRE].
  54. [54]
    L. Donnay, Asymptotic dynamics of three-dimensional gravity, PoS(Modave2015)001 [arXiv:1602.09021] [INSPIRE].
  55. [55]
    H. Afshar, S. Detournay, D. Grumiller and B. Oblak, Near-horizon geometry and warped conformal symmetry, JHEP 03 (2016) 187 [arXiv:1512.08233] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].CrossRefzbMATHGoogle Scholar
  57. [57]
    E. Kiritsis and C. Kounnas, String propagation in gravitational wave backgrounds, Phys. Lett. B 320 (1994) 264 [Addendum ibid. B 325 (1994) 536] [hep-th/9310202] [INSPIRE].
  58. [58]
    C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].ADSGoogle Scholar
  59. [59]
    C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].ADSGoogle Scholar
  60. [60]
    B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    A. Giveon and M. Roček, Supersymmetric string vacua on AdS 3 × N , JHEP 04 (1999) 019 [hep-th/9904024] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  62. [62]
    T. Harmark, J. Hartong and N.A. Obers, Nonrelativistic strings and limits of the AdS/CFT correspondence, Phys. Rev. D 96 (2017) 086019 [arXiv:1705.03535] [INSPIRE].ADSGoogle Scholar
  63. [63]
    J. Gomis, J. Gomis and K. Kamimura, Non-relativistic superstrings: a new soluble sector of AdS 5 × S 5, JHEP 12 (2005) 024 [hep-th/0507036] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    J.R. David, Anti-de Sitter gravity associated with the supergroup SU(1, 1|2) × SU(1, 1|2), Mod. Phys. Lett. A 14 (1999) 1143 [hep-th/9904068] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    J.M. Izquierdo and P.K. Townsend, Supersymmetric space-times in (2 + 1) AdS supergravity models, Class. Quant. Grav. 12 (1995) 895 [gr-qc/9501018] [INSPIRE].
  66. [66]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  67. [67]
    J.M. Maldacena and L. Maoz, Desingularization by rotation, JHEP 12 (2002) 055 [hep-th/0012025] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].ADSGoogle Scholar
  69. [69]
    O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
  70. [70]
    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].ADSCrossRefzbMATHGoogle Scholar
  71. [71]
    O. Lunin, S.D. Mathur and A. Saxena, What is the gravity dual of a chiral primary?, Nucl. Phys. B 655 (2003) 185 [hep-th/0211292] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  72. [72]
    I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  74. [74]
    I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  76. [76]
    M. Ammon, A. Castro and N. Iqbal, Wilson lines and entanglement entropy in higher spin gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  77. [77]
    J. de Boer and J.I. Jottar, Entanglement entropy and higher spin holography in AdS 3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].CrossRefGoogle Scholar
  78. [78]
    M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefanski and A. Torrielli, Protected string spectrum in AdS 3 /CFT 2 from worldsheet integrability, JHEP 04 (2017) 091 [arXiv:1701.03501] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  79. [79]
    M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].ADSGoogle Scholar
  80. [80]
    A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  81. [81]
    M. Henneaux and S.-J. Rey, Nonlinear W as asymptotic symmetry of three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  82. [82]
    M. Gary, D. Grumiller and R. Rashkov, Towards non-AdS holography in 3-dimensional higher spin gravity, JHEP 03 (2012) 022 [arXiv:1201.0013] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  83. [83]
    H. Afshar, M. Gary, D. Grumiller, R. Rashkov and M. Riegler, Non-AdS holography in 3-dimensional higher spin gravity — general recipe and example, JHEP 11 (2012) 099 [arXiv:1209.2860] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, Warped AdS 3 black holes, JHEP 03 (2009) 130 [arXiv:0807.3040] [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    D. Grumiller and N. Johansson, Consistent boundary conditions for cosmological topologically massive gravity at the chiral point, Int. J. Mod. Phys. D 17 (2009) 2367 [arXiv:0808.2575] [INSPIRE].ADSzbMATHGoogle Scholar
  86. [86]
    D. Grumiller, R. Jackiw and N. Johansson, Canonical analysis of cosmological topologically massive gravity at the chiral point, arXiv:0806.4185 [INSPIRE].
  87. [87]
    W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  88. [88]
    B. Chen and J. Long, High spin topologically massive gravity, JHEP 12 (2011) 114 [arXiv:1110.5113] [INSPIRE].ADSzbMATHGoogle Scholar
  89. [89]
    B. Chen, J. Long and J.-B. Wu, Spin-3 topologically massive gravity, Phys. Lett. B 705 (2011) 513 [arXiv:1106.5141] [INSPIRE].ADSCrossRefGoogle Scholar
  90. [90]
    E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional spin-3 theories based on general kinematical algebras, JHEP 01 (2017) 114 [arXiv:1612.02277] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Theoretical Physics and Delta Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingP.R. China
  3. 3.The Niels Bohr InstituteCopenhagen UniversityCopenhagen ØDenmark

Personalised recommendations