Journal of High Energy Physics

, 2018:90 | Cite as

Complexity and behind the horizon cut off

  • Amin Akhavan
  • Mohsen Alishahiha
  • Ali NasehEmail author
  • Hamed Zolfi
Open Access
Regular Article - Theoretical Physics


Motivated by \( T\overline{T} \) deformation of a conformal field theory we compute holographic complexity for a black brane solution with a cutoff using “complexity=action” proposal. In order to have a late time behavior consistent with Lloyd’s bound one is forced to have a cutoff behind the horizon whose value is fixed by the boundary cutoff. Using this result we compute holographic complexity for two dimensional AdS solutions where we get expected late times linear growth. It is in contrast with the naively computation which is done without assuming the cutoff where the complexity approaches a constant at the late time.


AdS-CFT Correspondence Black Holes Conformal Field Theory 


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]
    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
  2. [2]
    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].
  3. [3]
    S. Lloyd, Ultimate Physical limits to computation, Nature 406 (2000) 1047 [quant-ph/9908043].
  4. [4]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
  6. [6]
    F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
  7. [7]
    A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \) -deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
  8. [8]
    L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
  9. [9]
    M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
  10. [10]
    T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T 2 deformation,arXiv:1807.11401[INSPIRE].
  11. [11]
    M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Subregion Action and Complexity, arXiv:1809.06031 [INSPIRE].
  12. [12]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    K. Parattu, S. Chakraborty and T. Padmanabhan, Variational Principle for Gravity with Null and Non-null boundaries: A Unified Boundary Counter-term, Eur. Phys. J. C 76 (2016) 129 [arXiv:1602.07546] [INSPIRE].
  14. [14]
    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].
  15. [15]
    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, arXiv:1804.01561 [INSPIRE].
  16. [16]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Alishahiha, A. Faraji Astaneh, M.R. Mohammadi Mozaffar and A. Mollabashi, Complexity Growth with Lifshitz Scaling and Hyperscaling Violation, JHEP 07 (2018) 042 [arXiv:1802.06740] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Alishahiha, R. Fareghbal and A.E. Mosaffa, 2D Gravity on AdS 2 with Chern-Simons Corrections, JHEP 01 (2009) 069 [arXiv:0812.0453] [INSPIRE].
  19. [19]
    G. Guralnik, A. Iorio, R. Jackiw and S.Y. Pi, Dimensionally reduced gravitational Chern-Simons term and its kink, Annals Phys. 308 (2003) 222 [hep-th/0305117] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Grumiller and W. Kummer, The classical solutions of the dimensionally reduced gravitational Chern-Simons theory, Annals Phys. 308 (2003) 211 [hep-th/0306036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Alishahiha and F. Ardalan, Central Charge for 2D Gravity on AdS 2 and AdS 2 /CF T 1 Correspondence, JHEP 08 (2008) 079 [arXiv:0805.1861] [INSPIRE].
  23. [23]
    M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
  24. [24]
    S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  25. [25]
    A. Kitaev, A simple model of quantum holography (part 1), talks at KITP, April 7, 2015,
  26. [26]
    A. Kitaev, A simple model of quantum holography (part 2), talks at KITP, May 27, 2015,
  27. [27]
    K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
  28. [28]
    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].
  29. [29]
    J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
  30. [30]
    A.R. Brown, H. Gharibyan, H.W. Lin, L. Susskind, L. Thorlacius and Y. Zhao, The Case of the Missing Gates: Complexity of Jackiw-Teitelboim Gravity, arXiv:1810.08741 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Amin Akhavan
    • 1
  • Mohsen Alishahiha
    • 2
  • Ali Naseh
    • 1
    Email author
  • Hamed Zolfi
    • 3
  1. 1.School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.School of Physics, Institute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of PhysicsSharif University of TechnologyTehranIran

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