Renormalization group, secular term resummation and AdS (in)stability

  • Ben Craps
  • Oleg Evnin
  • Joris Vanhoof
Open Access


We revisit the issues of non-linear AdS stability, its relation to growing (secular) terms in na¨ıve perturbation theory around the AdS background, and the need and possible strategies for resumming such terms. To this end, we review a powerful and elegant resummation method, which is mathematically identical to the standard renormalization group treatment of ultraviolet divergences in perturbative quantum field theory. We apply this method to non-linear gravitational perturbation theory in the AdS background at first non-trivial order and display the detailed structure of the emerging renormalization flow equations. We prove, in particular, that a majority of secular terms (and the corresponding terms in the renormalization flow equations) that could be present on general grounds given the spectrum of frequencies of linear AdS perturbations, do not in fact arise.


Classical Theories of Gravity AdS-CFT Correspondence Holography and quark-gluon plasmas 


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]
    L.D. Landau and E..M. Lifshitz, Mechanics, 3rd edition, Reed Publishing, (1981), pg. 86.Google Scholar
  2. [2]
    J.D. Logan, Applied mathematics, 3rd edition, John Wiley & Sons, (2006), pg. 93.Google Scholar
  3. [3]
    L.-Y. Chen, N. Goldenfeld and Y. Oono, The renormalization group and singular perturbations: multiple scales, boundary layers and reductive perturbation theory, Phys. Rev. E 54 (1996) 376 [hep-th/9506161] [INSPIRE].ADSGoogle Scholar
  4. [4]
    V.I. Arnol’d, V.V. Kozlov and A.I. Neistadt, Mathematical aspects of classical and celestial mechanics, Springer, (1997).Google Scholar
  5. [5]
    Y. Nakayama, Holographic interpretation of renormalization group approach to singular perturbations in nonlinear differential equations, Phys. Rev. D 88 (2013) 105006 [arXiv:1305.4117] [INSPIRE].ADSGoogle Scholar
  6. [6]
    S. Kuperstein and A. Mukhopadhyay, Spacetime emergence via holographic RG flow from incompressible Navier-Stokes at the horizon, JHEP 11 (2013) 086 [arXiv:1307.1367] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    O.J.C. Dias, G.T. Horowitz and J.E. Santos, Gravitational turbulent instability of anti-de Sitter space, Class. Quant. Grav. 29 (2012) 194002 [arXiv:1109.1825] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    O.J.C. Dias, G.T. Horowitz, D. Marolf and J.E. Santos, On the nonlinear stability of asymptotically anti-de Sitter solutions, Class. Quant. Grav. 29 (2012) 235019 [arXiv:1208.5772] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Maliborski and A. Rostworowski, Time-periodic solutions in an Einstein AdS-massless-scalar-field system, Phys. Rev. Lett. 111 (2013) 051102 [arXiv:1303.3186] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Buchel, S.L. Liebling and L. Lehner, Boson stars in AdS spacetime, Phys. Rev. D 87 (2013) 123006 [arXiv:1304.4166] [INSPIRE].ADSGoogle Scholar
  12. [12]
    J. Abajo-Arrastia, E. da Silva, E. Lopez, J. Mas and A. Serantes, Holographic relaxation of finite size isolated quantum systems, JHEP 05 (2014) 126 [arXiv:1403.2632] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    M. Maliborski and A. Rostworowski, What drives AdS unstable?, Phys. Rev. D 89 (2014) 124006 [arXiv:1403.5434] [INSPIRE].ADSGoogle Scholar
  14. [14]
    V. Balasubramanian, A. Buchel, S.R. Green, L. Lehner and S.L. Liebling, Holographic thermalization, stability of AdS and the Fermi-Pasta-Ulam-Tsingou paradox, Phys. Rev. Lett. 113 (2014) 071601 [arXiv:1403.6471] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    J.A. Murdock, Perturbations: theory and methods, SIAM, (1987).Google Scholar
  16. [16]
    J. Shen, T. Tang and L.-L. Wang, Spectral methods: algorithms, analysis and applications, Springer, (2011).Google Scholar
  17. [17]
    J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Theoretische NatuurkundeVrije Universiteit Brussel and The International Solvay InstitutesBrusselsBelgium
  2. 2.Laboratoire de Physique Théorique, Ecole Normale SupérieureParis Cedex 05France
  3. 3.Department of Physics, Faculty of ScienceChulalongkorn UniversityBangkokThailand

Personalised recommendations