Gauge symmetries and holographic anomalies of Chern-Simons and transgression AdS gravity

Open Access
Regular Article - Theoretical Physics


We review the issue of gauge and gravitational anomalies with backgrounds, offering a new outlook on some aspects of these questions.

We compute the holographic anomalies of hypothetical theories dual, in the sense of the AdS-CFT correspondence, to Chern-Simons AdS gravity. Those anomalies are either gauge anomalies, associated to the AdS gauge group of the theory, or diffeomorphism anomalies, with each kind related to the other. AdS gauge anomalies include Weyl, Lorentz (gravitational) and gauge translations while diffeomorphism anomalies include gravitational and Weyl (or scale) anomalies. Our results therefore go beyond previous investigations on Chern-Simons AdS gravity holograpic anomalies, that dealt only with Weyl anomalies. Furthermore, our calculations were done allowing a non vanishing torsion, unlike previous works that considered only the zero torsion, or metric, case.

As a result of using suitable action principles for Chern-Simons AdS gravity, coming from Transgression forms, we obtain finite results without the need for further regularization.

Our results are of potential interest for Lovelock gravity theories, as it has been shown that the boundary terms dictated by the transgressions for Chern-Simons gravity are also suitable to regularize Lovelock theories. The Wess-Zumino consistency condition ensures that anomalies of the generic form computed here should appear for these and other theories.


AdS-CFT Correspondence Chern-Simons Theories Anomalies in Field and String Theories Field Theories in Higher Dimensions 


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. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A.H. Chamseddine, Topological gauge theory of gravity in five-dimensions and all odd dimensions, Phys. Lett. B 233 (1989) 291 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    A.H. Chamseddine, Topological gravity and supergravity in various dimensions, Nucl. Phys. B 346 (1990) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Bañados, R. Troncoso and J. Zanelli, Higher dimensional Chern-Simons supergravity, Phys. Rev. D 54 (1996) 2605 [gr-qc/9601003] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    R. Troncoso and J. Zanelli, New gauge supergravity in seven-dimensions and eleven-dimensions, Phys. Rev. D 58 (1998) 101703 [hep-th/9710180] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    J. Zanelli, Lecture notes on Chern-Simons (super-)gravities, hep-th/0502193 [INSPIRE].
  8. [8]
    R. Aros, M. Contreras,R. Olea, R. Troncoso and J. Zanelli, Charges in 2 + 1 Dimensional Gravity and Supergravity, presented at the Strings99 Conference, Potsdam Germany (1999).Google Scholar
  9. [9]
    P. Mora and H. Nishino, Fundamental extended objects for Chern-Simons supergravity, Phys. Lett. B 482 (2000) 222 [hep-th/0002077] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    P. Mora, Chern-Simons supersymmetric branes, Nucl. Phys. B 594 (2001) 229 [hep-th/0008180] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    A. Borowiec, M. Ferraris and M. Francaviglia, A covariant formalism for Chern-Simons gravity, J. Phys. A 36 (2003) 2589 [hep-th/0301146] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  12. [12]
    A. Borowiec, L. Fatibene, M. Ferraris and M. Francaviglia, Covariant Lagrangian formulation of Chern-Simons and BF theories, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 755 [hep-th/0511060] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    F. Izaurieta, E. Rodriguez and P. Salgado, On transgression forms and Chern-Simons (super)gravity, hep-th/0512014 [INSPIRE].
  14. [14]
    F. Izaurieta, E. Rodriguez and P. Salgado, The extended Cartan homotopy formula and a subspace separation method for Chern-Simons supergravity, Lett. Math. Phys. 80 (2007) 127 [hep-th/0603061] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    G. Sardanashvily, Gauge conservation laws in higher dimensional Chern-Simons models, hep-th/0303059 [INSPIRE].
  16. [16]
    G. Sardanashvily, Energy momentum conservation laws in higher dimensional Chern-Simons models, hep-th/0303148 [INSPIRE].
  17. [17]
    P. Mora, Transgressión forms as unifying principle in field theory, hep-th/0512255 [INSPIRE].
  18. [18]
    P. Mora, R. Olea, R. Troncoso and J. Zanelli, Finite action principle for Chern-Simons AdS gravity, JHEP 06 (2004) 036 [hep-th/0405267] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    P. Mora, R. Olea, R. Troncoso and J. Zanelli, Transgression forms and extensions of Chern-Simons gauge theories, JHEP 02 (2006) 067 [hep-th/0601081] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    B. Zumino, Y.-S. Wu and A. Zee, Chiral anomalies, higher dimensions and differential geometry, Nucl. Phys. B 239 (1984) 477 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    W.A. Bardeen and B. Zumino, Consistent and covariant anomalies in gauge and gravitational theories, Nucl. Phys. B 244 (1984) 421 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    J. Manes, R. Stora and B. Zumino, Algebraic study of chiral anomalies, Commun. Math. Phys. 102 (1985) 157 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    L. Alvarez-Gaumé and P. Ginsparg, The structure of gauge and gravitational anomalies, Annals Phys. 161 (1985) 423.ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  25. [25]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    S.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].ADSCrossRefMATHGoogle Scholar
  27. [27]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    S. Nojiri and S.D. Odintsov, On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence, Int. J. Mod. Phys. A 15 (2000) 413 [hep-th/9903033] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  30. [30]
    M. Blau, K.S. Narain and E. Gava, On subleading contributions to the AdS/CFT trace anomaly, JHEP 09 (1999) 018 [hep-th/9904179] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    A. Schwimmer and S. Theisen, Universal features of holographic anomalies, JHEP 10 (2003) 001 [hep-th/0309064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    M. Bañados, A. Schwimmer and S. Theisen, Chern-Simons gravity and holographic anomalies, JHEP 05 (2004) 039 [hep-th/0404245] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    M. Bañados, O. Mišković and S. Theisen, Holographic currents in first order gravity and finite Fefferman-Graham expansions, JHEP 06 (2006) 025 [hep-th/0604148] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Bañados, R. Olea and S. Theisen, Counterterms and dual holographic anomalies in CS gravity, JHEP 10 (2005) 067 [hep-th/0509179] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    N. Boulanger, General solutions of the Wess-Zumino consistency condition for the Weyl anomalies, JHEP 07 (2007) 069 [arXiv:0704.2472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    N. Boulanger, Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions, Phys. Rev. Lett. 98 (2007) 261302 [arXiv:0706.0340] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    P. Mora, Action Principles for Transgression and Chern-Simons AdS Gravities, JHEP 11 (2014) 128 [arXiv:1407.6032] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    C. Fefferman and R. Graham, Conformal Invariants, in The mathematical heritage of Elie Cartan Conference, Lyon France (1984), Astérisque Numero Hors Serie (1985) 95.Google Scholar
  40. [40]
    P. Mora, Transgressions and Holographic Conformal Anomalies for Chern-Simons Gravities, arXiv:1010.5110 [INSPIRE].
  41. [41]
    A. Das, B. Mukhopadhyaya and S. SenGupta, Why has spacetime torsion such negligible effect on the Universe?, Phys. Rev. D 90 (2014) 107901 [arXiv:1410.0814] [INSPIRE].ADSGoogle Scholar
  42. [42]
    R. Olea, Regularization of odd-dimensional AdS gravity: Kounterterms, JHEP 04 (2007) 073 [hep-th/0610230] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    I.G. Moss, Anomalies, boundaries and the in-in formalism, J. Phys. A 45 (2012) 374022 [arXiv:1201.5732] [INSPIRE].MathSciNetMATHGoogle Scholar
  44. [44]
    M. Nakahara, Geometry, Topology and Physics, IOP, Bristol U.K. (1991).Google Scholar
  45. [45]
    P. Mora, R. Olea, R. Troncoso and J. Zanelli, Vacuum energy in odd-dimensional AdS gravity, hep-th/0412046 [INSPIRE].
  46. [46]
    H.R. Afshar, Flat/AdS boundary conditions in three dimensional conformal gravity, JHEP 10 (2013) 027 [arXiv:1307.4855] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022 [hep-th/0508218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    S.N. Solodukhin, Holographic description of gravitational anomalies, JHEP 07 (2006) 003 [hep-th/0512216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Blagojevic, B. Cvetkovic, O. Mišković and R. Olea, Holography in 3D AdS gravity with torsion, JHEP 05 (2013) 103 [arXiv:1301.1237] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    O. Chandía and J. Zanelli, Topological invariants, instantons and chiral anomaly on spaces with torsion, Phys. Rev. D 55 (1997) 7580 [hep-th/9702025] [INSPIRE].ADSMathSciNetGoogle Scholar
  52. [52]
    O. Chandía and J. Zanelli, Supersymmetric particle in a space-time with torsion and the index theorem, Phys. Rev. D 58 (1998) 045014 [hep-th/9803034] [INSPIRE].ADSGoogle Scholar
  53. [53]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. 13 (1972) 874 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    B. Zumino, Gravity theories in more than four-dimensions, Phys. Rept. 137 (1986) 109 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    G. Kofinas and R. Olea, Universal regularization prescription for Lovelock AdS gravity, JHEP 11 (2007) 069 [arXiv:0708.0782] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Centro Universitario Regional Este (CURE)Universidad de la República, UruguayRochaUruguay

Personalised recommendations