Journal of High Energy Physics

, 2013:115 | Cite as

Holographic interpretations of the renormalization group

  • Vijay Balasubramanian
  • Monica Guica
  • Albion LawrenceEmail author
Open Access


In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension \( \varDelta \notin \frac{d}{2}+\mathbb{Z} \), for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action (S W ) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in S W as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.


Gauge-gravity correspondence AdS-CFT Correspondence Renormalization Group 


  1. [1]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  2. [2]
    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].MathSciNetADSGoogle Scholar
  3. [3]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
  4. [4]
    L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [INSPIRE].
  5. [5]
    E.T. Akhmedov, A remark on the AdS/CFT correspondence and the renormalization group flow, Phys. Lett. B 442 (1998) 152 [hep-th/9806217] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    V. Balasubramanian and P. Kraus, Space-time and the holographic renormalization group, Phys. Rev. Lett. 83 (1999) 3605 [hep-th/9903190] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. [8]
    M. Porrati and A. Starinets, RG fixed points in supergravity duals of 4D field theory and asymptotically AdS spaces, Phys. Lett. B 454 (1999) 77 [hep-th/9903085] [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: duality cascades and χSB resolution of naked singularities, JHEP 08 (2000) 052 [hep-th/0007191] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    D. Brattan, J. Camps, R. Loganayagam and M. Rangamani, CFT dual of the AdS Dirichlet problem : fluid/gravity on cut-off surfaces, JHEP 12 (2011) 090 [arXiv:1106.2577] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    D. Marolf and M. Rangamani, Causality and the AdS Dirichlet problem, JHEP 04 (2012) 035 [arXiv:1201.1233] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. [13]
    L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [14]
    H.L. Verlinde, Holography and compactification, Nucl. Phys. B 580 (2000) 264 [hep-th/9906182] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    I. Heemskerk and J. Polchinski, Holographic and Wilsonian renormalization groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].ADSGoogle Scholar
  18. [18]
    X. Dong, B. Horn, E. Silverstein and G. Torroba, Moduli stabilization and the holographic RG for AdS and DS, arXiv:1209.5392 [INSPIRE].
  19. [19]
    J. de Boer, The holographic renormalization group, Fortsch. Phys. 49 (2001) 339 [hep-th/0101026] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  20. [20]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. [22]
    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].ADSzbMATHCrossRefGoogle Scholar
  23. [23]
    M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in Anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].MathSciNetADSGoogle Scholar
  27. [27]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    O. Aharony, M. Berkooz and E. Silverstein, Multiple trace operators and nonlocal string theories, JHEP 08 (2001) 006 [hep-th/0105309] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  30. [30]
    M. Berkooz, A. Sever and A. Shomer, ’double tracedeformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A. Sever and A. Shomer, A note on multitrace deformations and AdS/CFT, JHEP 07 (2002) 027 [hep-th/0203168] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    A.W. Peet and J. Polchinski, UV/IR relations in AdS dynamics, Phys. Rev. D 59 (1999) 065011 [hep-th/9809022] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    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].ADSCrossRefGoogle Scholar
  34. [34]
    I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176 [INSPIRE].
  35. [35]
    A. Lawrence and A. Sever, Holography and renormalization in lorentzian signature, JHEP 10 (2006) 013 [hep-th/0606022] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    V. Balasubramanian, P. Kraus, A.E. Lawrence and S.P. Trivedi, Holographic probes of Anti-de Sitter space-times, Phys. Rev. D 59 (1999) 104021 [hep-th/9808017] [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    W. Muck, Running scaling dimensions in holographic renormalization group flows, JHEP 08 (2010) 085 [arXiv:1006.2987] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    E.T. Akhmedov, Notes on multitrace operators and holographic renormalization group, hep-th/0202055 [INSPIRE].
  39. [39]
    O. Aharony, M. Berkooz and E. Silverstein, Nonlocal string theories on AdS 3 × S 3 and stable nonsupersymmetric backgrounds, Phys. Rev. D 65 (2002) 106007 [hep-th/0112178] [INSPIRE].MathSciNetADSGoogle Scholar
  40. [40]
    M. Abramowitz and I. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Applied mathematics series, Dover Publications, Dover U.K. (1965).Google Scholar
  41. [41]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    N. Borodatchenkova, M. Haack and W. Muck, Towards holographic renormalization of fake supergravity, Nucl. Phys. B 815 (2009) 215 [arXiv:0811.3191] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    W. Mueck, An improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].ADSGoogle Scholar
  44. [44]
    T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT, JHEP 01 (2008) 019 [hep-th/0602106] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  46. [46]
    R. Feynman and J. Vernon, F.L., The theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24 (1963) 118 [INSPIRE].

Copyright information

© SISSA 2013

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
    • 2
  • Monica Guica
    • 1
  • Albion Lawrence
    • 3
    Email author
  1. 1.David Rittenhouse LaboratoryUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Laboratoire de Physique ThéoriqueÉcole Normale SupérieureParisFrance
  3. 3.Martin Fisher School of PhysicsBrandeis UniversityWalthamU.S.A.

Personalised recommendations