Journal of High Energy Physics

, 2019:102 | Cite as

Action growth of dyonic black holes and electromagnetic duality

  • Hai-Shan Liu
  • H. LüEmail author
Open Access
Regular Article - Theoretical Physics


Electromagnetic duality of Maxwell theory is a symmetry of equations but not of the action. The usual application of the “complexity = action” conjecture would thus lose this duality. It was recently proposed in arXivid:1901.00014 that the duality can be restored by adding some appropriate boundary term, at the price of introducing the mixed boundary condition in the variation principle. We present universal such a term in both first-order and second-order formalism for a general theory of a minimally-coupled Maxwell field. The first-order formalism has the advantage that the variation principle involves only the Dirichlet boundary condition. Including this term, we compute the on-shell actions in the Wheeler-De Witt patch and find that the duality is preserved in these actions for a variety of theories, including Einstein-Maxwell, Einstein-Maxwell-Dilaton, Einstein-Born-Infeld and Einstein-Horndeski-Maxwell theories.


Black Holes AdS-CFT Correspondence Gauge-gravity correspondence Classical Theories of Gravity 


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 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  2. [2]
    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].
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Condensed Matter Phys. 3 (2012) 9 [arXiv:1108.1197] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    J. McGreevy, TASI 2015 Lectures on Quantum Matter (with a View Toward Holographic Duality), in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., 1-26 June 2015, pp. 215-296 [] [arXiv:1606.08953] [INSPIRE].
  8. [8]
    J. Zaanen, Y.W. Sun, Y. Liu and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, Cambridge U.K. (2015) [INSPIRE].
  9. [9]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1402.5674] [INSPIRE].
  10. [10]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
  11. [11]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
  13. [13]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
  14. [14]
    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
  15. [15]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Y. Zhao, Complexity and Boost Symmetry, Phys. Rev. D 98 (2018) 086011 [arXiv:1702.03957] [INSPIRE].
  17. [17]
    S.-J. Zhang, Complexity and phase transitions in a holographic QCD model, Nucl. Phys. B 929 (2018) 243 [arXiv:1712.07583] [INSPIRE].
  18. [18]
    J. Jiang and H.-B. Zhang, Surface term, corner term and action growth in F (R abcd) gravity theory, Phys. Rev. D 99 (2019) 086005 [arXiv:1806.10312] [INSPIRE].
  19. [19]
    Z.-Y. Fan and M. Guo, Holographic complexity under a global quantum quench, arXiv:1811.01473 [INSPIRE].
  20. [20]
    J. Jiang, Action growth rate for a higher curvature gravitational theory, Phys. Rev. D 98 (2018) 086018 [arXiv:1810.00758] [INSPIRE].
  21. [21]
    S.A. Hosseini Mansoori, V. Jahnke, M.M. Qaemmaqami and Y.D. Olivas, Holographic complexity of anisotropic black branes, arXiv:1808.00067 [INSPIRE].
  22. [22]
    H. Ghaffarnejad, M. Farsam and E. Yaraie, Effects of quintessence dark energy on the action growth and butterfly velocity, arXiv:1806.05735 [INSPIRE].
  23. [23]
    E. Yaraie, H. Ghaffarnejad and M. Farsam, Complexity growth and shock wave geometry in AdS-Maxwell-power-Yang-Mills theory, Eur. Phys. J. C 78 (2018) 967 [arXiv:1806.07242] [INSPIRE].
  24. [24]
    Y.-S. An and R.-H. Peng, Effect of the dilaton on holographic complexity growth, Phys. Rev. D 97 (2018) 066022 [arXiv:1801.03638] [INSPIRE].
  25. [25]
    J. Jiang, Holographic complexity in charged Vaidya black hole, Eur. Phys. J. C 79 (2019) 130 [arXiv:1811.07347] [INSPIRE].
  26. [26]
    Seth Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047 [quant-ph/9908043].
  27. [27]
    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    H. Huang, X.-H. Feng and H. Lü, Holographic Complexity and Two Identities of Action Growth, Phys. Lett. B 769 (2017) 357 [arXiv:1611.02321] [INSPIRE].
  29. [29]
    W.-J. Pan and Y.-C. Huang, Holographic complexity and action growth in massive gravities, Phys. Rev. D 95 (2017) 126013 [arXiv:1612.03627] [INSPIRE].
  30. [30]
    M. Alishahiha, A. Faraji Astaneh, A. Naseh and M.H. Vahidinia, On complexity for F (R) and critical gravity, JHEP 05 (2017) 009 [arXiv:1702.06796] [INSPIRE].
  31. [31]
    P. Wang, H. Yang and S. Ying, Action growth in f (R) gravity, Phys. Rev. D 96 (2017) 046007 [arXiv:1703.10006] [INSPIRE].
  32. [32]
    W.-D. Guo, S.-W. Wei, Y.-Y. Li and Y.-X. Liu, Complexity growth rates for AdS black holes in massive gravity and f (R) gravity, Eur. Phys. J. C 77 (2017) 904 [arXiv:1703.10468] [INSPIRE].
  33. [33]
    P.A. Cano, R.A. Hennigar and H. Marrochio, Complexity Growth Rate in Lovelock Gravity, Phys. Rev. Lett. 121 (2018) 121602 [arXiv:1803.02795] [INSPIRE].
  34. [34]
    R.-G. Cai, M. Sasaki and S.-J. Wang, Action growth of charged black holes with a single horizon, Phys. Rev. D 95 (2017) 124002 [arXiv:1702.06766] [INSPIRE].
  35. [35]
    R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    X.-H. Feng and H.-S. Liu, Holographic Complexity Growth Rate in Horndeski Theory, Eur. Phys. J. C 79 (2019) 40 [arXiv:1811.03303] [INSPIRE].
  37. [37]
    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Z.-Y. Fan and M. Guo, On the Noether charge and the gravity duals of quantum complexity, JHEP 08 (2018) 031 [arXiv:1805.03796] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    J. Jiang and X.-W. Li, Modifiedcomplexity equals actionconjecture, arXiv:1903.05476 [INSPIRE].
  40. [40]
    D. Momeni, M. Faizal, S. Bahamonde and R. Myrzakulov, Holographic complexity for time-dependent backgrounds, Phys. Lett. B 762 (2016) 276 [arXiv:1610.01542] [INSPIRE].
  41. [41]
    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
  42. [42]
    Z.-Y. Fan and M. Guo, Holographic complexity and thermodynamics of AdS black holes, Phys. Rev. D 100 (2019) 026016 [arXiv:1903.04127] [INSPIRE].
  43. [43]
    A.R. Brown, H. Gharibyan, H.W. Lin, L. Susskind, L. Thorlacius and Y. Zhao, Complexity of Jackiw-Teitelboim gravity, Phys. Rev. D 99 (2019) 046016 [arXiv:1810.08741] [INSPIRE].
  44. [44]
    K. Goto, H. Marrochio, R.C. Myers, L. Queimada and B. Yoshida, Holographic Complexity Equals Which Action?, JHEP 02 (2019) 160 [arXiv:1901.00014] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Deser and C. Teitelboim, Duality Transformations of Abelian and Nonabelian Gauge Fields, Phys. Rev. D 13 (1976) 1592 [INSPIRE].
  46. [46]
    S. Deser, Off-Shell Electromagnetic Duality Invariance, J. Phys. A 15 (1982) 1053 [INSPIRE].
  47. [47]
    S. Deser, M. Henneaux and C. Teitelboim, Electric-magnetic black hole duality, Phys. Rev. D 55 (1997) 826 [hep-th/9607182] [INSPIRE].
  48. [48]
    E. Cremmer, B. Julia, H. Lü and C.N. Pope, Dualization of dualities. 2. Twisted self-duality of doubled fields and superdualities, Nucl. Phys. B 535 (1998) 242 [hep-th/9806106] [INSPIRE].
  49. [49]
    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
  50. [50]
    H.W. Braden, J.D. Brown, B.F. Whiting and J.W. York Jr., Charged black hole in a grand canonical ensemble, Phys. Rev. D 42 (1990) 3376 [INSPIRE].
  51. [51]
    S.W. Hawking and S.F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52 (1995) 5865 [hep-th/9504019] [INSPIRE].
  52. [52]
    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
  53. [53]
    C.J. Gao and S.N. Zhang, Dilaton black holes in de Sitter or Anti-de Sitter universe, Phys. Rev. D 70 (2004) 124019 [hep-th/0411104] [INSPIRE].
  54. [54]
    H. Lü, Charged dilatonic AdS black holes and magnetic AdS D−2 × R 2 vacua, JHEP 09 (2013) 112 [arXiv:1306.2386] [INSPIRE].
  55. [55]
    M. Born and L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. Lond. A 144 (1934) 425 [INSPIRE].
  56. [56]
    A. García D, H. Salazar I and J.F. Plebanski, Type-D solutions of the Einstein and Born-Infeld nonlinear electrodynamics equations, Nuovo Cim. B 84 (1984) 65.Google Scholar
  57. [57]
    S. Li, H. Lü and H. Wei, Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory in Diverse Dimensions, JHEP 07 (2016) 004 [arXiv:1606.02733] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  59. [59]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    A. Anabalon, A. Cisterna and J. Oliva, Asymptotically locally AdS and flat black holes in Horndeski theory, Phys. Rev. D 89 (2014) 084050 [arXiv:1312.3597] [INSPIRE].
  61. [61]
    A. Cisterna and C. Erices, Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario, Phys. Rev. D 89 (2014) 084038 [arXiv:1401.4479] [INSPIRE].
  62. [62]
    X.-H. Feng, H.-S. Liu, H. Lü and C.N. Pope, Black Hole Entropy and Viscosity Bound in Horndeski Gravity, JHEP 11 (2015) 176 [arXiv:1509.07142] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    X.-H. Feng, H.-S. Liu, H. Lü and C.N. Pope, Thermodynamics of Charged Black Holes in Einstein-Horndeski-Maxwell Theory, Phys. Rev. D 93 (2016) 044030 [arXiv:1512.02659] [INSPIRE].
  64. [64]
    J. Beltran Jimenez, R. Durrer, L. Heisenberg and M. Thorsrud, Stability of Horndeski vector-tensor interactions, JCAP 10 (2013) 064 [arXiv:1308.1867] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    T. Kobayashi, H. Motohashi and T. Suyama, Black hole perturbation in the most general scalar-tensor theory with second-order field equations II: the even-parity sector, Phys. Rev. D 89 (2014) 084042 [arXiv:1402.6740] [INSPIRE].
  66. [66]
    M. Minamitsuji, Causal structure in the scalar-tensor theory with field derivative coupling to the Einstein tensor, Phys. Lett. B 743 (2015) 272 [INSPIRE].
  67. [67]
    X.-M. Kuang and E. Papantonopoulos, Building a Holographic Superconductor with a Scalar Field Coupled Kinematically to Einstein Tensor, JHEP 08 (2016) 161 [arXiv:1607.04928] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    W.-J. Jiang, H.-S. Liu, H. Lü and C.N. Pope, DC Conductivities with Momentum Dissipation in Horndeski Theories, JHEP 07 (2017) 084 [arXiv:1703.00922] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    M. Baggioli and W.-J. Li, Diffusivities bounds and chaos in holographic Horndeski theories, JHEP 07 (2017) 055 [arXiv:1705.01766] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    H.-S. Liu, H. Lü and C.N. Pope, Holographic Heat Current as Noether Current, JHEP 09 (2017) 146 [arXiv:1708.02329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    X.-H. Feng, H.-S. Liu, W.-T. Lu and H. Lü, Horndeski Gravity and the Violation of Reverse Isoperimetric Inequality, Eur. Phys. J. C 77 (2017) 790 [arXiv:1705.08970] [INSPIRE].
  72. [72]
    E. Caceres, R. Mohan and P.H. Nguyen, On holographic entanglement entropy of Horndeski black holes, JHEP 10 (2017) 145 [arXiv:1707.06322] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    W.-J. Geng, S.-L. Li, H. Lü and H. Wei, Gödel metrics with chronology protection in Horndeski gravities, Phys. Lett. B 780 (2018) 196 [arXiv:1801.00009] [INSPIRE].
  74. [74]
    Y.-Z. Li and H. Lü, a-theorem for Horndeski gravity at the critical point, Phys. Rev. D 97 (2018) 126008 [arXiv:1803.08088] [INSPIRE].
  75. [75]
    H.-S. Liu, Violation of Thermal Conductivity Bound in Horndeski Theory, Phys. Rev. D 98 (2018) 061902 [arXiv:1804.06502] [INSPIRE].
  76. [76]
    Y.-Z. Li, H. Lü and H.-Y. Zhang, Scale Invariance vs. Conformal Invariance: Holographic Two-Point Functions in Horndeski Gravity, Eur. Phys. J. C 79 (2019) 592 [arXiv:1812.05123] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Joint Quantum Studies and Department of Physics, School of ScienceTianjin UniversityTianjinChina
  2. 2.Institute for Advanced Physics & MathematicsZhejiang University of TechnologyHangzhouChina

Personalised recommendations