Skip to main content
SpringerLink
Log in

Lovelock theory and the AdS/CFT correspondence

  • Review Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

Lovelock theory is the natural extension of general relativity to higher dimensions. It can be also thought of as a toy model for ghost-free higher curvature corrections in gravitational theories. It admits a family of AdS vacua, which provides an appealing arena to explore different holographic aspects in a broader setup within the context of the AdS/CFT correspondence. We will elaborate on these features and review previous work concerning the constraints that Lovelock theory entails on the CFT parameters when imposing conditions like unitarity, positivity of the energy or causality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Parallel to this is the fact that Lovelock gravity might not be a consistent low energy truncation of any point in the moduli space of a putative UV complete (such as, for instance, M-) theory. In that respect, the relevance of these vacua is not a priori guaranteed.

  2. Many important features of the static spherically symmetric solutions of Lovelock gravities were already understood in the late eighties [42–46], greatly contributing to the acceptance of these theories as physically relevant. Subsequent work exploring in detail the case of degenerate Lovelock theory, i.e., when the gravitational couplings are such that there is a unique (A)dS vacuum, have been pursued in [47, 48]; see also [49–51]. The reader may also want to consult [52, 53] for a nice recent report on the subject.

  3. This latter expression was earlier found by Jacobson and Myers in the Hamiltonian approach to black hole entropy [71], and justified a short time ago in [72], following the lines of reasoning of [67]. The connection with the Iyer-Wald formula was further explored very recently in [73].

  4. A different and interesting approach has been pursued in [80].

References

  1. Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys. 12, 498 (1971)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Zwiebach, B.: Curvature squared terms and string theories. Phys. Lett. B 156, 315 (1985)

    Article  ADS  Google Scholar 

  3. Zumino, B.: Gravity theories in more than four dimensions. Phys. Rep. 137, 109 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  4. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)

    ADS  MATH  MathSciNet  Google Scholar 

  5. Padmanabhan, T., Kothawala, D.: Lanczos-Lovelock models of gravity. arXiv:1302.2151 [gr-qc]

  6. Edelstein, J.D.: Lovelock theory, black holes and holography. In: García-Parrado, A., Mena. F.C., Moura, F., Vaz, E. (eds.) Progress in Mathematical Relativity, Gravitation and Cosmology. Springer Proceedings in Mathematics & Statistics, vol. 60, p. 19. Springer, Berlin (2013)

  7. Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravitation, gauge theories and differential geometry. Phys. Rep. 66, 213 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  8. Gravanis, E., Willison, S.: Intersecting membranes in AdS and Lovelock gravity. J. Math. Phys. 47, 092503 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  9. Zanelli, J.: Lecture Notes on Chern-Simons (super-)gravities. hep-th/0502193

  10. Regge, T.: On broken symmetries and gravity. Phys. Rep. 137, 31 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  11. Lanczos, C.: A remarkable property of the Riemann-Christoffel tensor in four dimensions. Ann. Math. 39, 842 (1938)

    Article  MathSciNet  Google Scholar 

  12. Madore, J.: Kaluza-Klein theory with the Lanczos lagrangian. Phys. Lett. A 110, 289 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  13. Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  14. Myers, R.: Higher-derivative gravity, surface terms, and string theory. Phys. Rev. D 36, 392 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  15. Israel, W.: Singular hypersurfaces and thin shells in general relativity. Nuovo Cimento B 44, 1 (1966) [ Erratum-ibid. B 48 (1967) 463]

    Google Scholar 

  16. Gravanis, E., Willison, S.: Israel conditions for the Gauss-Bonnet theory and the Friedmann equation on the brane universe. Phys. Lett. B 562, 118 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Camanho, X.O., Edelstein, J.D., Giribet, G., Gomberoff, A.: New type of phase transition in gravitational theories. Phys. Rev. D 86, 124048 (2012)

    Article  ADS  Google Scholar 

  18. Camanho, X. O., Edelstein, J. D., Giribet, G., Gomberoff, A.: Generalized phase transitions in Lovelock gravity. arXiv:1311.6768 [hep-th]

  19. Boulware, D.G., Deser, S.: String generated gravity models. Phys. Rev. Lett. 55, 2656 (1985)

    Article  ADS  Google Scholar 

  20. Horowitz, G.T., Itzhaki, N.: Black holes, shock waves, and causality in the AdS/CFT correspondence. JHEP 9902, 010 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  21. Hofman, D.M., Maldacena, J.: Conformal collider physics: energy and charge correlations. JHEP 0805, 012 (2008)

    Article  ADS  Google Scholar 

  22. Aharony, O., Gubser, S.S., Maldacena, J.M., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rep. 323, 183 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  23. Edelstein, J.D., Portugues, R.: Gauge/string duality in confining theories. Fortschr. Phys. 54, 525 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Casalderrey-Solana, J., Liu, H., Mateos, D., Rajagopal, K., Wiedemann, U.A.: Gauge/string duality, hot QCD and heavy ion collisions. arXiv:1101.0618 [hep-th]

  25. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  26. Witten, E.: Anti-de sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998)

    ADS  MATH  MathSciNet  Google Scholar 

  27. Osborn, H., Petkou, A.C.: Implications of conformal invariance in field theories for general dimensions. Ann. Phys. 231, 311 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Buchel, A., Escobedo, J., Myers, R.C., Paulos, M.F., Sinha, A., Smolkin, M.: Holographic GB gravity in arbitrary dimensions. JHEP 1003, 111 (2010)

    Article  ADS  Google Scholar 

  29. Camanho, X.O., Edelstein, J.D., Paulos, M.F.: Lovelock theories, holography and the fate of the viscosity bound. JHEP 1105, 127 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  30. Arutyunov, G., Frolov, S.: Three point Green function of the stress energy tensor in the AdS/CFT correspondence. Phys. Rev. D 60, 026004 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  31. Liu, H., Tseytlin, A.A.: D=4 super Yang-Mills, D=5 gauged supergravity, and D=4 conformal supergravity. Nucl. Phys. B 533, 88 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Erdmenger, J., Osborn, H.: Conserved currents and the energy-momentum tensor in conformally invariant theories for general dimensions. Nucl. Phys. B 483, 431 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Hofman, D.M.: Higher derivative gravity, causality and positivity of energy in a UV complete QFT. Nucl. Phys. B 823, 174 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Zhiboedov, A.: On conformal field theories with extremal a/c values. arXiv:1304.6075 [hep-th]

  35. Kulaxizi, M., Parnachev, A.: Supersymmetry constraints in holographic gravities. Phys. Rev. D 82, 066001 (2010)

    Article  ADS  Google Scholar 

  36. de Boer, J., Kulaxizi, M., Parnachev, A.: Holographic Lovelock gravities and black holes. JHEP 1006, 008 (2010)

    Article  Google Scholar 

  37. Camanho, X.O., Edelstein, J.D.: Causality in AdS/CFT and Lovelock theory. JHEP 1006, 099 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  38. Ozkan, M., Pang, Y.: Supersymmetric completion of Gauss-Bonnet combination in five dimensions. JHEP 1303, 158 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  39. Camanho, X.O., Edelstein, J.D.: Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity. JHEP 1004, 007 (2010)

    Article  ADS  Google Scholar 

  40. Buchel, A., Myers, R.C.: Causality of holographic hydrodynamics. JHEP 0908, 016 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. de Boer, J., Kulaxizi, M., Parnachev, A.: \(\text{ AdS }_7/\text{ CFT }_6\), Gauss-Bonnet gravity, and viscosity bound. JHEP 1003, 087 (2010)

    Article  Google Scholar 

  42. Wheeler, J.T.: Symmetric solutions to the Gauss-Bonnet extended Einstein equations. Nucl. Phys. B 268, 737 (1986)

    Article  ADS  Google Scholar 

  43. Wheeler, J.T.: Symmetric solutions to the maximally Gauss-Bonnet extended Einstein equations. Nucl. Phys. B 273, 732 (1986)

    Article  ADS  MATH  Google Scholar 

  44. Myers, R.C., Simon, J.Z.: Black hole thermodynamics in Lovelock gravity. Phys. Rev. D 38, 2434 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  45. Wiltshire, D.L.: Black holes in string generated gravity models. Phys. Rev. D 38, 2445 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  46. Whitt, B.: Spherically symmetric solutions of general second order gravity. Phys. Rev. D 38, 3000 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  47. Bañados, M., Teitelboim, C., Zanelli, J.: Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem. Phys. Rev. Lett. 72, 957 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. Crisostomo, J., Troncoso, R., Zanelli, J.: Black hole scan. Phys. Rev. D 62, 084013 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  49. Cai, R.-G., Soh, K.-S.: Topological black holes in the dimensionally continued gravity. Phys. Rev. D 59, 044013 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  50. Cai, R.-G.: Gauss-Bonnet black holes in AdS spaces. Phys. Rev. D 65, 084014 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  51. Dadhich, N., Pons, J.M., Prabhu, K.: On the static Lovelock black holes. Gen. Relativ. Gravit. 45, 1131 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  52. Charmousis, C.: Higher order gravity theories and their black hole solutions. In: Lect. Notes Phys., vol. 769, p. 299 (2009)

  53. Garraffo, C., Giribet, G.: The Lovelock black holes. Mod. Phys. Lett. A 23, 1801 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. Camanho, X.O., Edelstein, J.D.: A Lovelock black hole bestiary. Class. Quantum Gravity 30, 035009 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  55. Brigante, M., Liu, H., Myers, R.C., Shenker, S., Yaida, S.: Viscosity bound violation in higher derivative gravity. Phys. Rev. D 77, 126006 (2008)

    Article  ADS  Google Scholar 

  56. Brigante, M., Liu, H., Myers, R.C., Shenker, S., Yaida, S.: The viscosity bound and causality violation. Phys. Rev. Lett. 100, 191601 (2008)

    Article  ADS  Google Scholar 

  57. Horowitz, G.T., Itzhaki, N.: Black holes, shock waves, and causality in the AdS/CFT correspondence. JHEP 9902, 010 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  58. Lang, R.: Propagation of gravitons in the shock wave geometry. B.Sc. Thesis, MIT, (2009) http://hdl.handle.net/1721.1/51580

  59. Nojiri, S’i, Odintsov, S.D.: On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence. Int. J. Mod. Phys. A 15, 413 (2000)

    ADS  MATH  MathSciNet  Google Scholar 

  60. Kulaxizi, M., Parnachev, A.: Energy flux positivity and unitarity in CFTs. Phys. Rev. Lett. 106, 011601 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  61. Camanho, X.O., Edelstein, J.D.: Cosmic censorship in Lovelock theory. JHEP 1311, 151 (2013)

    Google Scholar 

  62. Zamolodchikov, A.B.: Irreversibility of the flux of the renormalization group in a 2D field theory. JETP Lett. 43, 730 (1986) [ Pisma, Zh. Eksp. Teor. Fiz. 43 (1986) 565]

  63. Jack, I., Osborn, H.: Analogs for the c-theorem for four-dimensional renormalizable field theories. Nucl. Phys. B 343, 647 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  64. Myers, R.C., Sinha, A.: Holographic c-theorems in arbitrary dimensions. JHEP 1101, 125 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  65. Komargodski, Z., Schwimmer, A.: On renormalization group flows in four dimensions. JHEP 1112, 099 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  66. Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  67. Lewkowycz, A., Maldacena, J.: Generalized gravitational, entropy. JHEP 1308, 090 (2013)

    Google Scholar 

  68. Hung, L.-Y., Myers, R.C., Smolkin, M.: On holographic entanglement entropy and higher curvature gravity. JHEP 1104, 025 (2011)

    Article  ADS  Google Scholar 

  69. Iyer, V., Wald, R.M.: Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D 50, 846 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  70. de Boer, J., Kulaxizi, M., Parnachev, A.: Holographic entanglement entropy in Lovelock gravities. JHEP 1107, 109 (2011)

    Article  ADS  Google Scholar 

  71. Jacobson, T., Myers, R.C.: Black hole entropy and higher curvature interactions. Phys. Rev. Lett. 70, 3684 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  72. Bhattacharyya, A., Kaviraj, A., Sinha, A.: Entanglement entropy in higher derivative holography. JHEP 1308, 012 (2013)

    Article  ADS  Google Scholar 

  73. Bhattacharyya, A., Sharma, M., Sinha, A.: On generalized gravitational entropy, squashed cones and holography. arXiv:1308.5748 [hep-th]

  74. Hubeny, V.E., Minwalla, S., Rangamani, M.: The fluid/gravity correspondence. arXiv:1107.5780 [hep-th]

  75. Cremonini, S.: The shear viscosity to entropy ratio: a status report. Mod. Phys. Lett. B 25, 1867 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  76. Buchel, A.: Shear viscosity of CFT plasma at finite coupling. Phys. Lett. B 665, 298 (2008)

    Article  ADS  Google Scholar 

  77. Buchel, A., Cremonini, S.: Viscosity bound and causality in superfluid plasma. JHEP 1010, 026 (2010)

    Article  ADS  Google Scholar 

  78. Buchel, A., Heller, M.P., Myers, R.C.: sQGP as hCFT. Phys. Lett. B 680, 521 (2009)

    Article  ADS  Google Scholar 

  79. Buchel, A., Myers, R.C., Sinha, A.: Beyond \(\eta /s = 1/4\pi \). JHEP 0903, 084 (2009)

    Article  ADS  Google Scholar 

  80. Hu, Y.-P., Li, H.-F., Nie, Z.-Y.: The first order hydrodynamics via AdS/CFT correspondence in the Gauss-Bonnet gravity. JHEP 1101, 123 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  81. Kovtun, P., Son, D.T., Starinets, A.O.: Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett. 94, 111601 (2005)

    Article  ADS  Google Scholar 

  82. Shu, F.-W.: The quantum viscosity bound in Lovelock gravity. Phys. Lett. B 685, 325 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  83. Myers, R.C., Paulos, M.F., Sinha, A.: Holographic studies of quasi-topological gravity. JHEP 1008, 035 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  84. Bhattacharya, J., Bhattacharyya, S., Minwalla, S., Yarom, A.: A theory of first order dissipative superfluid, dynamics. arXiv:1105.3733 [hep-th]

  85. Minwalla, S.: The entropy current in hydrodynamics, superfluid hydrodynamics and gravity, talk at Strings (2011)

Download references

Acknowledgments

We wish to thank Alex Buchel, Gastón Giribet, Andy Gomberoff, Diego Hofman, Manuela Kulaxizi, Juan Maldacena, Rob Myers, Miguel Paulos and Sasha Zhiboedov for discussions on these subjects held over the last few years. This work was supported in part by MICINN and FEDER (grant FPA2011-22594), by Xunta de Galicia (Consellería de Educación and grant PGIDIT10PXIB206075PR), and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). X.O.C. is thankful to the Front of Galician-speaking Scientists for encouragement.

Author information

Authors and Affiliations

  1. Department of Particle Physics and IGFAE, University of Santiago de Compostela, 15782 , Santiago de Compostela, Spain

    Xián O. Camanho, José D. Edelstein & José M. Sánchez de Santos

  2. Centro de Estudios Científicos (CECs), Valdivia, Chile

    José D. Edelstein

Authors
  1. Xián O. Camanho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José D. Edelstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José M. Sánchez de Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José D. Edelstein.

Additional information

This article belongs to the Topical Collection: Progress in Mathematical Relativity with Applications to Astrophysics and Cosmology.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Camanho, X.O., Edelstein, J.D. & Sánchez de Santos, J.M. Lovelock theory and the AdS/CFT correspondence. Gen Relativ Gravit 46, 1637 (2014). https://doi.org/10.1007/s10714-013-1637-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-013-1637-3

Keywords

Search

Navigation