WdW-patches in AdS3 and complexity change under conformal transformations II

  • Mario FloryEmail author
Open Access
Regular Article - Theoretical Physics


We study the null-boundaries of Wheeler-de Witt (WdW) patches in three dimensional Poincaré-AdS, when the selected boundary timeslice is an arbitrary (non-constant) function, presenting some useful analytic statements about them. Special attention will be given to the piecewise smooth nature of the null-boundaries, due to the emergence of caustics and null-null joint curves. This is then applied, in the spirit of one of our previous papers, to the problem of how the complexity of the CFT2 groundstate changes under a small local conformal transformation according to the action (CA) proposal. In stark contrast to the volume (CV) proposal, where this change is only proportional to the second order in the infinitesimal expansion parameter σ, we show that in the CA case we obtain terms of order σ and even σ log(σ). This has strong implications for the possible field-theory duals of the CA proposal, ruling out an entire class of them.


AdS-CFT Correspondence Gauge-gravity correspondence 


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]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
  2. [2]
    M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum Computation as Geometry, Science 311 (2006) 1133 [quant-ph/0603161].
  3. [3]
    D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
  4. [4]
    L. Susskind, Butterflies on the Stretched Horizon, arXiv:1311.7379 [INSPIRE].
  5. [5]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    L. Susskind, Three Lectures on Complexity and Black Holes, 2018, arXiv:1810.11563 [INSPIRE].
  7. [7]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
  9. [9]
    L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
  10. [10]
    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].ADSGoogle Scholar
  11. [11]
    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].
  12. [12]
    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  14. [14]
    R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].
  15. [15]
    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    T. Ali, A. Bhattacharyya, S. Shajidul Haque, E.H. Kim and N. Moynihan, Time Evolution of Complexity: A Critique of Three Methods, JHEP 04 (2019) 087 [arXiv:1810.02734] [INSPIRE].ADSGoogle Scholar
  17. [17]
    S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [arXiv:1810.05151] [INSPIRE].ADSGoogle Scholar
  18. [18]
    J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].
  20. [20]
    P. Caputa and J.M. Magan, Quantum Computation as Gravity, arXiv:1807.04422 [INSPIRE].
  21. [21]
    A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
  24. [24]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].
  26. [26]
    A. Bhattacharyya, P. Caputa, S.R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-Integral Complexity for Perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C 79 (2019) 109 [arXiv:1803.01797] [INSPIRE].
  28. [28]
    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, More on complexity of operators in quantum field theory, JHEP 03 (2019) 161 [arXiv:1809.06678] [INSPIRE].ADSGoogle Scholar
  29. [29]
    H.W. Lin, Cayley graphs and complexity geometry, JHEP 02 (2019) 063 [arXiv:1808.06620] [INSPIRE].ADSGoogle Scholar
  30. [30]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    K. Hashimoto, N. Iizuka and S. Sugishita, Time evolution of complexity in Abelian gauge theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].
  32. [32]
    K. Hashimoto, N. Iizuka and S. Sugishita, Thoughts on Holographic Complexity and its Basis-dependence, Phys. Rev. D 98 (2018) 046002 [arXiv:1805.04226] [INSPIRE].
  33. [33]
    R. Abt et al., Topological Complexity in AdS 3 /CFT 2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
  34. [34]
    R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].ADSzbMATHGoogle Scholar
  35. [35]
    R.-Q. Yang and K.-Y. Kim, Complexity of operators generated by quantum mechanical Hamiltonians, JHEP 03 (2019) 010 [arXiv:1810.09405] [INSPIRE].ADSGoogle Scholar
  36. [36]
    A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].
  37. [37]
    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].ADSMathSciNetzbMATHGoogle Scholar
  38. [38]
    Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, JHEP 02 (2019) 145 [arXiv:1804.01561] [INSPIRE].ADSzbMATHGoogle Scholar
  40. [40]
    M. Flory and N. Miekley, Complexity change under conformal transformations in AdS 3 /CFT 2, JHEP 05 (2019) 003 [arXiv:1806.08376] [INSPIRE].
  41. [41]
    T. Numasawa, Holographic Complexity for disentangled states, arXiv:1811.03597 [INSPIRE].
  42. [42]
    A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a new York time story, JHEP 03 (2019) 044 [arXiv:1811.03097] [INSPIRE].ADSzbMATHGoogle Scholar
  43. [43]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].ADSGoogle Scholar
  44. [44]
    M. Moosa and I. Shehzad, Is volume the holographic dual of fidelity susceptibility?, arXiv:1809.10169 [INSPIRE].
  45. [45]
    G. Mandal, R. Sinha and N. Sorokhaibam, The inside outs of AdS 3 /CFT 2 : exact AdS wormholes with entangled CFT duals, JHEP 01 (2015) 036 [arXiv:1405.6695] [INSPIRE].
  46. [46]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  47. [47]
    R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    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].
  49. [49]
    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    D. Hilbert, Die Grundlagen der Physik. 1, Gott. Nachr. 27 (1915) 395.Google Scholar
  51. [51]
    A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, Berlin Germany (1917), pg. 142.Google Scholar
  52. [52]
    J.W. York, Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082.ADSGoogle Scholar
  53. [53]
    G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752.Google Scholar
  54. [54]
    G. Hayward, Gravitational action for spacetimes with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275.Google Scholar
  55. [55]
    D. Brill and G. Hayward, Is the gravitational action additive?, Phys. Rev. D 50 (1994) 4914 [gr-qc/9403018] [INSPIRE].
  56. [56]
    K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, A Boundary Term for the Gravitational Action with Null Boundaries, Gen. Rel. Grav. 48 (2016) 94 [arXiv:1501.01053] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  57. [57]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  58. [58]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
  59. [59]
    R.-Q. Yang, Strong energy condition and complexity growth bound in holography, Phys. Rev. D 95 (2017) 086017 [arXiv:1610.05090] [INSPIRE].
  60. [60]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
  61. [61]
    I. Jubb, J. Samuel, R. Sorkin and S. Surya, Boundary and Corner Terms in the Action for General Relativity, Class. Quant. Grav. 34 (2017) 065006 [arXiv:1612.00149] [INSPIRE].
  62. [62]
    M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Subregion Action and Complexity, arXiv:1809.06031 [INSPIRE].
  63. [63]
    M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  64. [64]
    V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  65. [65]
    M.M. Sheikh-Jabbari and H. Yavartanoo, On quantization of AdS 3 gravity I: semi-classical analysis, JHEP 07 (2014) 104 [arXiv:1404.4472] [INSPIRE].
  66. [66]
    M.M. Sheikh-Jabbari and H. Yavartanoo, On 3d bulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locally AdS 3 geometries, Eur. Phys. J. C 76 (2016) 493 [arXiv:1603.05272] [INSPIRE].
  67. [67]
    M.M. Sheikh-Jabbari and H. Yavartanoo, Excitation entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 94 (2016) 126006 [arXiv:1605.00341] [INSPIRE].
  68. [68]
    M.J. Bhaseen, B. Doyon, A. Lucas and K. Schalm, Far from equilibrium energy flow in quantum critical systems, arXiv:1311.3655 [INSPIRE].
  69. [69]
    J. Erdmenger, D. Fernandez, M. Flory, E. Megias, A.-K. Straub and P. Witkowski, Time evolution of entanglement for holographic steady state formation, JHEP 10 (2017) 034 [arXiv:1705.04696] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  70. [70]
    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
  71. [71]
    M. Moosa, Divergences in the rate of complexification, Phys. Rev. D 97 (2018) 106016 [arXiv:1712.07137] [INSPIRE].
  72. [72]
    D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  73. [73]
    C. Akers, R. Bousso, I.F. Halpern and G.N. Remmen, Boundary of the future of a surface, Phys. Rev. D 97 (2018) 024018 [arXiv:1711.06689] [INSPIRE].
  74. [74]
    Z.-Y. Fan and M. Guo, Holographic complexity under a global quantum quench, arXiv:1811.01473 [INSPIRE].
  75. [75]
    S. Chapman, D. Ge and G. Policastro, Holographic Complexity for Defects Distinguishes Action from Volume, JHEP 05 (2019) 049 [arXiv:1811.12549] [INSPIRE].Google Scholar
  76. [76]
    M. Flory, A complexity/fidelity susceptibility g-theorem for AdS 3 /BCFT 2, JHEP 06 (2017) 131 [arXiv:1702.06386] [INSPIRE].
  77. [77]
    T. Takayanagi, Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [arXiv:1808.09072] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  78. [78]
    E. Caceres and M.-L. Xiao, Complexity-action of subregions with corners, JHEP 03 (2019) 062 [arXiv:1809.09356] [INSPIRE].ADSGoogle Scholar
  79. [79]
    E. Bakhshaei and A. Shirzad, Entanglement entropy and complexity of singular subregions in deformed CFT, arXiv:1902.02504 [INSPIRE].
  80. [80]
    C.V. Johnson, D-branes, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2005),
  81. [81]
    E. Gourgoulhon and J.L. Jaramillo, A 3 + 1 perspective on null hypersurfaces and isolated horizons, Phys. Rept. 423 (2006) 159 [gr-qc/0503113] [INSPIRE].
  82. [82]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetGoogle Scholar
  83. [83]
    E. Witten, Light Rays, Singularities and All That, arXiv:1901.03928 [INSPIRE].
  84. [84]
    V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  85. [85]
    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
  86. [86]
    V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].
  87. [87]
    B. Czech and L. Lamprou, Holographic definition of points and distances, Phys. Rev. D 90 (2014) 106005 [arXiv:1409.4473] [INSPIRE].
  88. [88]
    R.C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].ADSGoogle Scholar
  89. [89]
    M. Headrick, R.C. Myers and J. Wien, Holograsphic Holes and Differential Entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].ADSzbMATHGoogle Scholar
  90. [90]
    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute of PhysicsJagiellonian UniversityKrakówPoland

Personalised recommendations