Advertisement

Journal of High Energy Physics

, 2019:220 | Cite as

Towards the generalized gravitational entropy for spacetimes with non-Lorentz invariant duals

  • Qiang WenEmail author
Open Access
Regular Article - Theoretical Physics
  • 18 Downloads

Abstract

Based on the Lewkowycz-Maldacena prescription and the fine structure analysis of holographic entanglement proposed in [1], we explicitly calculate the holographic entanglement entropy for warped CFT that duals to AdS3 with a Dirichlet-Neumann type of boundary conditions. We find that certain type of null geodesics emanating from the entangling surface ∂\( \mathcal{A} \) relate the field theory UV cutoff and the gravity IR cutoff. Inspired by the construction, we furthermore propose an intrinsic prescription to calculate the generalized gravitational entropy for general spacetimes with non-Lorentz invariant duals. Compared with the RT formula, there are two main differences. Firstly, instead of requiring that the bulk extremal surface ℰ should be anchored on ∂\( \mathcal{A} \), we require the consistency between the boundary and bulk causal structures to determine the corresponding ℰ. Secondly we use the null geodesics (or hypersurfaces) emanating from ∂\( \mathcal{A} \) and normal to ℰ to regulate ℰ in the bulk. We apply this prescription to flat space in three dimensions and get the entanglement entropies straightforwardly.

Keywords

Conformal Field Theory Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) 

Notes

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.

References

  1. [1]
    Q. Wen, Fine structure in holographic entanglement and entanglement contour, Phys. Rev. D 98 (2018) 106004 [arXiv:1803.05552] [INSPIRE].ADSGoogle Scholar
  2. [2]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. Song, Q. Wen and J. Xu, Modifications to holographic entanglement entropy in warped CFT, JHEP 02 (2017) 067 [arXiv:1610.00727] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    H. Jiang, W. Song and Q. Wen, Entanglement entropy in flat holography, JHEP 07 (2017) 142 [arXiv:1706.07552] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    W. Song, Q. Wen and J. Xu, Generalized gravitational entropy for warped Anti-de Sitter space, Phys. Rev. Lett. 117 (2016) 011602 [arXiv:1601.02634] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    X. Dong and A. Lewkowycz, Entropy, extremality, Euclidean variations and the equations of motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [INSPIRE].
  16. [16]
    M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    M. Headrick, General properties of holographic entanglement entropy, JHEP 03 (2014) 085 [arXiv:1312.6717] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    F.M. Haehl et al., Topological aspects of generalized gravitational entropy, JHEP 05 (2015) 023 [arXiv:1412.7561] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    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
  27. [27]
    G. Compère, M. Guica and M.J. Rodriguez, Two Virasoro symmetries in stringy warped AdS 3, JHEP 12 (2014) 012 [arXiv:1407.7871] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    S. Detournay, T. Hartman and D.M. Hofman, Warped conformal field theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [INSPIRE].ADSGoogle Scholar
  29. [29]
    W. Song and J. Xu, Correlation functions of warped CFT, JHEP 04 (2018) 067 [arXiv:1706.07621] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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. A269 (1962) 21.Google Scholar
  31. [31]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103.Google Scholar
  32. [32]
    A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Bagchi, Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Bagchi and R. Fareghbal, BMS/GCA redux: towards flatspace holography from non-relativistic symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    G. Compère, W. Song and A. Strominger, New boundary conditions for AdS 3, JHEP 05 (2013) 152 [arXiv:1303.2662] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Castro, D.M. Hofman and N. Iqbal, Entanglement entropy in warped conformal field theories, JHEP 02 (2016) 033 [arXiv:1511.00707] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    T. Azeyanagi, S. Detournay and M. Riegler, Warped black holes in lower-spin gravity, arXiv:1801.07263 [INSPIRE].
  39. [39]
    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].ADSCrossRefGoogle Scholar
  40. [40]
    R. Basu and M. Riegler, Wilson lines and holographic entanglement entropy in galilean conformal field theories, Phys. Rev. D 93 (2016) 045003 [arXiv:1511.08662] [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    E. Hijano and C. Rabideau, Holographic entanglement and Poincaré blocks in three-dimensional flat space, JHEP 05 (2018) 068 [arXiv:1712.07131] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    M. Asadi and R. Fareghbal, Holographic calculation of BMSFT mutual and 3-partite information, Eur. Phys. J. C 78 (2018) 620 [arXiv:1802.06618] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    R. Fareghbal and P. Karimi, Complexity growth in flat spacetimes, Phys. Rev. D 98 (2018) 046003 [arXiv:1806.07273] [INSPIRE].ADSGoogle Scholar
  44. [44]
    R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    G. Compere and S. Detournay, Boundary conditions for spacelike and timelike warped AdS 3 spaces in topologically massive gravity, JHEP 08 (2009) 092 [arXiv:0906.1243] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic diagrams, gravitational interactions & OPE structures, JHEP 06 (2017) 099 [arXiv:1702.06128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    G. Compère, W. Song and A. Strominger, Chiral Liouville gravity, JHEP 05 (2013) 154 [arXiv:1303.2660] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    D.M. Hofman and B. Rollier, Warped conformal field theory as lower spin gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    K. Jensen, Locality and anomalies in warped conformal field theory, JHEP 12 (2017) 111 [arXiv:1710.11626] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    L. Apolo and W. Song, Bootstrapping holographic warped CFTs or: how I learned to stop worrying and tolerate negative norms, JHEP 07 (2018) 112 [arXiv:1804.10525] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    R.D. Sorkin, Development of simplectic methods for the metrical and electromagnetic fields, Ph.D. thesis, Caltech, Pasadena, U.S.A. (1974)Google Scholar
  53. [53]
    Y. Neiman, The imaginary part of the gravity action and black hole entropy, JHEP 04 (2013) 071 [arXiv:1301.7041] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Ann. Phys. 140 (1982) 372.ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    S.A. Gentle and C. Keeler, On the reconstruction of Lifshitz spacetimes, JHEP 03 (2016) 195 [arXiv:1512.04538] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    T. Azeyanagi, W. Li and T. Takayanagi, On string theory duals of Lifshitz-like fixed points, JHEP 06 (2009) 084 [arXiv:0905.0688] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    S.N. Solodukhin, Entanglement entropy in non-relativistic field theories, JHEP 04 (2010) 101 [arXiv:0909.0277] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    V. Keranen, E. Keski-Vakkuri and L. Thorlacius, Thermalization and entanglement following a non-relativistic holographic quench, Phys. Rev. D 85 (2012) 026005 [arXiv:1110.5035] [INSPIRE].ADSGoogle Scholar
  60. [60]
    B.S. Kim, Schrödinger Holography with and without Hyperscaling Violation, JHEP 06 (2012) 116 [arXiv:1202.6062] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    M. Alishahiha, A. Faraji Astaneh and M.R. Mohammadi Mozaffar, Thermalization in backgrounds with hyperscaling violating factor, Phys. Rev. D 90 (2014) 046004 [arXiv:1401.2807] [INSPIRE].ADSGoogle Scholar
  62. [62]
    P. Fonda et al., Holographic thermalization with Lifshitz scaling and hyperscaling violation, JHEP 08 (2014) 051 [arXiv:1401.6088] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    S. Fischetti and D. Marolf, Complex entangling surfaces for AdS and Lifshitz black holes?, Class. Quant. Grav. 31 (2014) 214005 [arXiv:1407.2900] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    S.M. Hosseini and I. Véliz-Osorio, Entanglement and mutual information in two-dimensional nonrelativistic field theories, Phys. Rev. D 93 (2016) 026010 [arXiv:1510.03876] [INSPIRE].ADSMathSciNetGoogle Scholar
  65. [65]
    M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement in Lifshitz-type quantum field theories, JHEP 07 (2017) 120 [arXiv:1705.00483] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    T. He, J.M. Magan and S. Vandoren, Entanglement entropy in Lifshitz theories, SciPost Phys. 3 (2017) 034 [arXiv:1705.01147] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    S.A. Gentle and S. Vandoren, Lifshitz entanglement entropy from holographic cMERA, JHEP 07 (2018) 013 [arXiv:1711.11509] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement evolution in Lifshitz-type scalar theories, JHEP 01 (2019) 137 [arXiv:1811.11470] [INSPIRE].CrossRefzbMATHGoogle Scholar
  69. [69]
    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
  70. [70]
    J. Cheyne and D. Mattingly, Constructing entanglement wedges for Lifshitz spacetimes with Lifshitz gravity, Phys. Rev. D 97 (2018) 066024 [arXiv:1707.05913] [INSPIRE].ADSMathSciNetGoogle Scholar
  71. [71]
    A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
  72. [72]
    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS bootstrap, Class. Quant. Grav. 34 (2017) 174002 [arXiv:1705.05890] [INSPIRE].
  74. [74]
    E. Hijano, Semi-classical BMS 3 blocks and flat holography, JHEP 10 (2018) 044 [arXiv:1805.00949] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  75. [75]
    Y. Chen and G. Vidal, Entanglement contour J. Stat. Mech. 10 (2014) 10011 [arXiv:1406.1471].MathSciNetCrossRefGoogle Scholar
  76. [76]
    A. Botero and B. Reznik, Spatial structures and localization of vacuum entanglement in the linear harmonic chain, Phys. Rev. A 70 (2004) 052329.ADSCrossRefGoogle Scholar
  77. [77]
    I. Frérot and T. Roscilde, Area law and its violation: a microscopic inspection into the structure of entanglement and fluctuations, Phys. Rev. B 92 (2015) 115129.ADSCrossRefGoogle Scholar
  78. [78]
    A. Coser, C. De Nobili and E. Tonni, A contour for the entanglement entropies in harmonic lattices, J. Phys. A 50 (2017) 314001 [arXiv:1701.08427] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  79. [79]
    E. Tonni, J. Rodríguez-Laguna and G. Sierra, Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems, J. Stat. Mech. 1804 (2018) 043105 [arXiv:1712.03557] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Shing-Tung Yau Center and School of MathematicsSoutheast UniversityNanjingChina
  2. 2.Graduate SchoolChina Academy of Engineering PhysicsBeijingChina

Personalised recommendations