Journal of High Energy Physics

, 2018:26 | Cite as

4D spin-2 fields from 5D Chern-Simons theory

  • N. L. González AlbornozEmail author
  • D. Lüst
  • S. Salgado
  • A. Schmidt-May
Open Access
Regular Article - Theoretical Physics


We consider a 5-dimensional Chern-Simons gauge theory for the isometry group of Anti-de-Sitter spacetime, AdS4+1 ≃ SO(4, 2), and invoke different dimensional reduction schemes in order to relate it to 4-dimensional spin-2 theories. The AdS gauge algebra is isomorphic to a parametrized 4-dimensional conformal algebra, and the gauge fields corresponding to the generators of non-Abelian translations and special conformal transformations reduce to two vierbein fields in D = 4. Besides these two vierbeine, our reduction schemes leave only the Lorentz spin connection as an additional dynamical field in the 4-dimensional theories. We identify the corresponding actions as particular generalizations of Einstein-Cartan theory, conformal gravity and ghost-free bimetric gravity in first-order form.


Chern-Simons Theories Classical Theories of Gravity 


Open Access

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  1. [1]
    J. Zanelli, Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), in Proceedings, 7th Mexican Workshop on Particles and Fields (MWPF 1999), Merida, Mexico, November 10–17, 1999 (2005) [hep-th/0502193] [INSPIRE].
  2. [2]
    J. Zanelli, Uses of Chern-Simons actions, AIP Conf. Proc. 1031 (2008) 115 [arXiv:0805.1778] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Zeitschr. 9 (1921) 110.CrossRefzbMATHGoogle Scholar
  4. [4]
    M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
  7. [7]
    M. Ostrogradsky Mémoires sur les équations différentielles, relatives au problème des isopérimètres, Mem. Ac. St. Petersbourg 14 (1850) 385.Google Scholar
  8. [8]
    S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].ADSGoogle Scholar
  10. [10]
    A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49 (2016) 183001 [arXiv:1512.00021] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    S.F. Hassan, A. Schmidt-May and M. von Strauss, Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory, Universe 1 (2015) 92 [arXiv:1303.6940] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    S.F. Hassan, A. Schmidt-May and M. von Strauss, Extended Weyl Invariance in a Bimetric Model and Partial Masslessness, Class. Quant. Grav. 33 (2016) 015011 [arXiv:1507.06540] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Gording and A. Schmidt-May, Ghost-free infinite derivative gravity, JHEP 09 (2018) 044 [Erratum ibid. 10 (2018) 115] [arXiv:1807.05011] [INSPIRE].
  14. [14]
    J.D. Edelstein, M. Hassaine, R. Troncoso and J. Zanelli, Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert Lagrangian, Phys. Lett. B 640 (2006) 278 [hep-th/0605174] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    F. Izaurieta, E. Rodriguez, P. Minning, P. Salgado and A. Perez, Standard General Relativity from Chern-Simons Gravity, Phys. Lett. B 678 (2009) 213 [arXiv:0905.2187] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    R. Aros, M. Romo and N. Zamorano, Compactification in first order gravity, J. Phys. Conf. Ser. 134 (2008) 012013 [arXiv:0705.1162] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    I. Morales, B. Neves, Z. Oporto and O. Piguet, Chern-Simons gravity in four dimensions, Eur. Phys. J. C 77 (2017) 87 [arXiv:1701.03642] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A.H. Chamseddine, Topological gravity and supergravity in various dimensions, Nucl. Phys. B 346 (1990) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Bañados, Dilaton gravity (with a Gauss-Bonnet term) derived from five-dimensional Chern-Simons gravity, Phys. Rev. D 55 (1997) 2051 [gr-qc/9603029] [INSPIRE].
  20. [20]
    H. Nastase, D = 4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction, hep-th/0703034 [INSPIRE].
  21. [21]
    F. Izaurieta and E. Rodriguez, Effectively four-dimensional spacetimes emerging from d = 5 Einstein-Gauss-Bonnet Gravity, Class. Quant. Grav. 30 (2013) 155009 [arXiv:1207.1496] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    R. Aros and D.E. Diaz, AdS Chern-Simons Gravity induces Conformal Gravity, Phys. Rev. D 89 (2014) 084026 [arXiv:1311.5364] [INSPIRE].ADSGoogle Scholar
  23. [23]
    I. Morales, B. Neves, Z. Oporto and O. Piguet, Dimensionally compactified Chern-Simon theory in 5D as a gravitation theory in 4D, Int. J. Mod. Phys. Conf. Ser. 45 (2017) 1760005 [arXiv:1612.00409] [INSPIRE].CrossRefGoogle Scholar
  24. [24]
    M. Bañados, L.J. Garay and M. Henneaux, The Local degrees of freedom of higher dimensional pure Chern-Simons theories, Phys. Rev. D 53 (1996) 593 [hep-th/9506187] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    M. Bañados, L.J. Garay and M. Henneaux, The Dynamical structure of higher dimensional Chern-Simons theory, Nucl. Phys. B 476 (1996) 611 [hep-th/9605159] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Bañados, M. Henneaux, C. Iannuzzo and C.M. Viallet, A Note on the gauge symmetries of pure Chern-Simons theories with p form gauge fields, Class. Quant. Grav. 14 (1997) 2455 [gr-qc/9703061] [INSPIRE].
  27. [27]
    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
  28. [28]
    C. Lanczos, Elektromagnetismus als natürliche Eigenschaft der Riemannschen Geometrie, Z. Phys. 73 (1932) 147.ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    C. Lanczos, A remarkable property of the Riemann-Christoffel tensor in four dimensions, Annals Math. 39 (1938) 842 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R. Troncoso and J. Zanelli, Higher dimensional gravity, propagating torsion and AdS gauge invariance, Class. Quant. Grav. 17 (2000) 4451 [hep-th/9907109] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  33. [33]
    A.H. Chamseddine, Topological Gauge Theory of Gravity in Five-dimensions and All Odd Dimensions, Phys. Lett. B 233 (1989) 291 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    M. Bañados, R. Troncoso and J. Zanelli, Higher dimensional Chern-Simons supergravity, Phys. Rev. D 54 (1996) 2605 [gr-qc/9601003] [INSPIRE].
  35. [35]
    K. Hinterbichler and R.A. Rosen, Interacting Spin-2 Fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S.F. Hassan, A. Schmidt-May and M. von Strauss, On Partially Massless Bimetric Gravity, Phys. Lett. B 726 (2013) 834 [arXiv:1208.1797] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Apolo and S.F. Hassan, Non-linear partially massless symmetry in an SO(1, 5) continuation of conformal gravity, Class. Quant. Grav. 34 (2017) 105005 [arXiv:1609.09514] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    M. Ammon and J. Erdmenger, Gauge/gravity duality: foundations and applications, Cambridge University Press (2015) [INSPIRE].
  39. [39]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    N. Rahmanpour, N. Khosravi and B. Vakili, SO(4, 2) and derivatively coupled dRGT massive gravity, J. Geom. Phys. 135 (2019) 106 [arXiv:1801.10412] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    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

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Arnold Sommerfeld Center for Theoretical PhysicsLudwig-Maximilians-Universität MünchenMunichGermany
  2. 2.Max-Planck-Institut für PhysikMunichGermany

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