A practical approach to the Hamilton-Jacobi formulation of holographic renormalization

  • Henriette Elvang
  • Marios Hadjiantonis
Open Access
Regular Article - Theoretical Physics


We revisit the subject of holographic renormalization for asymptotically AdS spacetimes. For many applications of holography, one has to handle the divergences associated with the on-shell gravitational action. The brute force approach uses the Fefferman- Graham (FG) expansion near the AdS boundary to identify the divergences, but subsequent reversal of the expansion is needed to construct the infinite counterterms. While in principle straightforward, the method is cumbersome and application/reversal of FG is formally unsatisfactory. Various authors have proposed an alternative method based on the Hamilton-Jacobi equation. However, this approach may appear to be abstract, difficult to implement, and in some cases limited in applicability. In this paper, we clarify the Hamilton-Jacobi formulation of holographic renormalization and present a simple algorithm for its implementation to extract cleanly the infinite counterterms. While the derivation of the method relies on the Hamiltonian formulation of general relativity, the actual application of our algorithm does not. The work applies to any D-dimensional holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and arbitrary slicing. We illustrate the method in several examples, including the FGPW model, a holographic model of 3d ABJM theory, and cases with marginal scalars such as a dilaton-axion system.


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]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  5. [5]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C. Fefferman and C. Robin Graham, Conformal invariants, in Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque, France (1985), pg. 95.Google Scholar
  7. [7]
    J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    E.P. Verlinde and H.L. Verlinde, RG flow, gravity and the cosmological constant, JHEP 05 (2000) 034 [hep-th/9912018] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    E.P. Verlinde, On RG flow and the cosmological constant, Class. Quant. Grav. 17 (2000) 1277 [hep-th/9912058] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    J. de Boer, The holographic renormalization group, Fortsch. Phys. 49 (2001) 339 [hep-th/0101026] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Kalkkinen, D. Martelli and W. Mueck, Holographic renormalization and anomalies, JHEP 04 (2001) 036 [hep-th/0103111] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    D. Martelli and W. Mueck, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys. B 654 (2003) 248 [hep-th/0205061] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].MathSciNetMATHGoogle Scholar
  14. [14]
    I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP 11 (2010) 014 [arXiv:1007.4592] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    I. Papadimitriou, Holographic renormalization of general dilaton-axion gravity, JHEP 08 (2011) 119 [arXiv:1106.4826] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    F. Larsen and R. McNees, Inflation and de Sitter holography, JHEP 07 (2003) 051 [hep-th/0307026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    D.Z. Freedman and S.S. Pufu, The holography of F -maximization, JHEP 03 (2014) 135 [arXiv:1302.7310] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    N. Bobev, H. Elvang, U. Kol, T. Olson and S.S. Pufu, Holography for N = 1 on S 4, arXiv:1605.00656 [INSPIRE].
  21. [21]
    H. Goldstein, Classical mechanics, 2nd edition, Addison-Wesley, U.S.A. July 1980 [ISBN-13:978-0201029185].Google Scholar
  22. [22]
    W. Mueck, Correlation functions in holographic renormalization group flows, Nucl. Phys. B 620 (2002) 477 [hep-th/0105270] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 super Yang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    N. Bobev, H. Elvang, D.Z. Freedman and S.S. Pufu, Holography for N = 2 on S 4, JHEP 07 (2014) 001 [arXiv:1311.1508] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of Physics and Michigan Center for Theoretical PhysicsUniversity of MichiganAnn ArborU.S.A.

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