Journal of High Energy Physics

, 2019:1 | Cite as

Holographic OPE coefficients from AdS black holes with matters

  • Yue-Zhou Li
  • Zhan-Feng MaiEmail author
  • H. Lü
Open Access
Regular Article - Theoretical Physics


We study the OPE coefficients cΔ, J for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension ΔL in an AdS black hole. We verify that the OPE coefficient cd,0 = 0 for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer ΔL, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients γΔ,J of the log(z\( \overline{z} \)) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for γΔ,J. We then consider charged AdS black holes in gauged supergravity STU models in D = 5 and D = 7, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with ΔL = d − 2. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient cd−2,0. Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how cd,0 can be nonzero with exchanging scalar operators in these backgrounds.


AdS-CFT Correspondence Conformal Field Theory 


Open Access

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  1. [1]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  2. [2]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle 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]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Henningson and K. Skenderis, Holography and the weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N. Banerjee and S. Dutta, Shear viscosity to entropy density ratio in six derivative gravity, JHEP 07 (2009) 024 [arXiv:0903.3925] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Renyi entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    Y.-Z. Li, Holographic studies of the generic massless cubic gravities, Phys. Rev. D 99 (2019) 066014 [arXiv:1901.03349] [INSPIRE].ADSGoogle Scholar
  11. [11]
    Y.-Z. Li, H. Lü and Z.-F. Mai, Universal structure of covariant holographic two-point functions in massless higher-order gravities, JHEP 10 (2018) 063 [arXiv:1808.00494] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P. Bueno, P.A. Cano, R.A. Hennigar and R.B. Mann, Universality of squashed-sphere partition functions, Phys. Rev. Lett. 122 (2019) 071602 [arXiv:1808.02052] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    H. Lü and R. Wen, Holographic (a, c)-charges and their universal relation in d = 6 from massless higher-order gravities, Phys. Rev. D 99 (2019) 126003 [arXiv:1901.11037] [INSPIRE].ADSGoogle Scholar
  14. [14]
    J.D. Qualls, Lectures on conformal field theory, arXiv:1511.04074 [INSPIRE].
  15. [15]
    S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, arXiv:1601.05000.
  16. [16]
    D. Simmons-Duffin, The conformal bootstrap, arXiv:1602.07982 [INSPIRE].
  17. [17]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
  18. [18]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].ADSzbMATHGoogle Scholar
  19. [19]
    K.B. Alkalaev and V.A. Belavin, Monodromic vs. geodesic computation of Virasoro classical conformal blocks, Nucl. Phys. B 904 (2016) 367 [arXiv:1510.06685] [INSPIRE].
  20. [20]
    A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic diagrams, gravitational interactions & OPE structures, JHEP 06 (2017) 099 [arXiv:1702.06128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    E. Dyer, D.Z. Freedman and J. Sully, Spinning geodesic Witten diagrams, JHEP 11 (2017) 060 [arXiv:1702.06139] [INSPIRE].
  22. [22]
    H.-Y. Chen, E.-J. Kuo and H. Kyono, Anatomy of geodesic Witten diagrams, JHEP 05 (2017) 070 [arXiv:1702.08818] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Kraus et al., Witten diagrams for torus conformal blocks, JHEP 09 (2017) 149 [arXiv:1706.00047] [INSPIRE].
  24. [24]
    A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from holography, arXiv:1903.05306 [INSPIRE].
  25. [25]
    D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783 [hep-th/9712074] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].ADSGoogle Scholar
  30. [30]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A. Belin, D.M. Hofman and G. Mathys, Einstein gravity from ANEC correlators, JHEP 08 (2019) 032 [arXiv:1904.05892] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, superconvergence and a stringy equivalence principle, arXiv:1904.05905 [INSPIRE].
  36. [36]
    J. Oliva and S. Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoffs theorem and C-function, Class. Quant. Grav. 27 (2010) 225002 [arXiv:1003.4773] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035 [arXiv:1004.2055] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    R.C. Myers and B. Robinson, Black holes in quasi-topological gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Y.-Z. Li, H.-S. Liu and H. Lü, Quasi-topological Ricci polynomial gravities, JHEP 02 (2018) 166 [arXiv:1708.07198] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    J. Peng and X.-H. Feng, Holographic aspects of quasi-topological gravity, arXiv:1802.00697 [INSPIRE].
  41. [41]
    P. Bueno and P.A. Cano, Einsteinian cubic gravity, Phys. Rev. D 94 (2016) 104005 [arXiv:1607.06463] [INSPIRE].
  42. [42]
    R.A. Hennigar and R.B. Mann, Black holes in Einsteinian cubic gravity, Phys. Rev. D 95 (2017) 064055 [arXiv:1610.06675] [INSPIRE].ADSMathSciNetGoogle Scholar
  43. [43]
    P. Bueno and P.A. Cano, Four-dimensional black holes in Einsteinian cubic gravity, Phys. Rev. D 94 (2016) 124051 [arXiv:1610.08019] [INSPIRE].
  44. [44]
    P. Bueno, P.A. Cano and A. Ruipérez, Holographic studies of Einsteinian cubic gravity, JHEP 03 (2018) 150 [arXiv:1802.00018] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M. Kulaxizi, G.S. Ng and A. Parnachev, Black holes, heavy states, phase shift and anomalous dimensions, SciPost Phys. 6 (2019) 065 [arXiv:1812.03120] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadi’c, Black holes and conformal Regge bootstrap, arXiv:1904.00060 [INSPIRE].
  47. [47]
    R.C. Myers, M.F. Paulos and A. Sinha, Holographic hydrodynamics with a chemical potential, JHEP 06 (2009) 006 [arXiv:0903.2834] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    X.-H. Feng and H. Lü, Higher-derivative gravity with non-minimally coupled Maxwell field, Eur. Phys. J. C 76 (2016) 178 [arXiv:1512.09153] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
  50. [50]
    M.J. Duff, J.T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality, Nucl. Phys. B 459 (1996) 125 [hep-th/9508094] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    K. Behrndt, M. Cvetič and W.A. Sabra, Nonextreme black holes of five-dimensional N = 2 AdS supergravity, Nucl. Phys. B 553 (1999) 317 [hep-th/9810227] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  52. [52]
    D.D.K. Chow, Single-rotation two-charge black holes in gauged supergravity, arXiv:1108.5139 [INSPIRE].
  53. [53]
    A. Anabalon, Exact Black Holes and Universality in the Backreaction of non-linear σ-models with a potential in (A)dS4, JHEP 06 (2012) 127 [arXiv:1204.2720] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    A. Anabalon, D. Astefanesei and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, JHEP 10 (2013) 184 [arXiv:1308.1693] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    A. Anabalón and D. Astefanesei, On attractor mechanism of AdS 4 black holes, Phys. Lett. B 727 (2013) 568 [arXiv:1309.5863] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    P.A. González, E. Papantonopoulos, J. Saavedra and Y. Vásquez, Four-dimensional asymptotically ads black holes with scalar hair, JHEP 12 (2013) 021 [arXiv:1309.2161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    X.-H. Feng, H. Lü and Q. Wen, Scalar hairy black holes in general dimensions, Phys. Rev. D 89 (2014) 044014 [arXiv:1312.5374] [INSPIRE].ADSGoogle Scholar
  58. [58]
    Z.-Y. Fan and H. Lü, Charged black holes with scalar hair, JHEP 09 (2015) 060 [arXiv:1507.04369] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    X. Zhang and H. Lü, Exact black hole formation in asymptotically (A) dS and flat spacetimes, Phys. Lett. B 736 (2014) 455 [arXiv:1403.6874] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    H. Lü and X. Zhang, Exact collapse solutions in D = 4, \( \mathcal{N} \) = 4 gauged supergravity and their generalizations, JHEP 07 (2014) 099 [arXiv:1404.7603] [INSPIRE].CrossRefGoogle Scholar
  61. [61]
    W. Xu, Exact black hole formation in three dimensions, Phys. Lett. B 738 (2014) 472 [arXiv:1409.3368] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    Z.-Y. Fan and H. Lü, Static and dynamic hairy planar black holes, Phys. Rev. D 92 (2015) 064008 [arXiv:1505.03557] [INSPIRE].
  63. [63]
    Z.-Y. Fan and B. Chen, Exact formation of hairy planar black holes, Phys. Rev. D 93 (2016) 084013 [arXiv:1512.09145] [INSPIRE].
  64. [64]
    H. Lü, C.N. Pope and Q. Wen, Thermodynamics of AdS black holes in Einstein-Scalar gravity, JHEP 03 (2015) 165 [arXiv:1408.1514] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    H.-S. Liu, H. Lü and C.N. Pope, Generalized Smarr formula and the viscosity bound for Einstein-Maxwell-dilaton black holes, Phys. Rev. D 92 (2015) 064014 [arXiv:1507.02294] [INSPIRE].
  66. [66]
    H.-S. Liu and H. Lü, Scalar charges in asymptotic AdS geometries, Phys. Lett. B 730 (2014) 267 [arXiv:1401.0010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    D.D.K. Chow and G. Compère, Dyonic AdS black holes in maximal gauged supergravity, Phys. Rev. D 89 (2014) 065003 [arXiv:1311.1204] [INSPIRE].
  69. [69]
    M.S. Bremer et al., Instanton cosmology and domain walls from M-theory and string theory, Nucl. Phys. B 543 (1999) 321 [hep-th/9807051] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Joint Quantum Studies and Department of Physics, School of ScienceTianjin UniversityTianjinChina

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