Tensor Galileons and gravity

  • Athanasios ChatzistavrakidisEmail author
  • Fech Scen Khoo
  • Diederik Roest
  • Peter Schupp
Open Access
Regular Article - Theoretical Physics


The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian p-forms. In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixed-symmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to p-forms and scalars with a shift-symmetry. The simplest case corresponds to linearised gravity with Lovelock invariants, relating the Galileon symmetry to diffeomorphisms. Finally, we examine the coupling of a mixed-symmetry tensor to gravity, and demonstrate in an explicit example that the inclusion of appropriate counterterms retains second order field equations.


Classical Theories of Gravity Gauge Symmetry Global Symmetries 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]
    M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres (in French), Mem. Acad. St. Petersburg VI 4 (1850) 385 [INSPIRE].
  2. [2]
    M. Zumalacárregui and J. García-Bellido, Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian, Phys. Rev. D 89 (2014) 064046 [arXiv:1308.4685] [INSPIRE].
  3. [3]
    J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Healthy theories beyond Horndeski, Phys. Rev. Lett. 114 (2015) 211101 [arXiv:1404.6495] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    C. Deffayet, G. Esposito-Farese and D.A. Steer, Counting the degrees of freedom of generalized galileons, Phys. Rev. D 92 (2015) 084013 [arXiv:1506.01974] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, JHEP 12 (2016) 100 [arXiv:1608.08135] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi and D. Langlois, Healthy degenerate theories with higher derivatives, JCAP 07 (2016) 033 [arXiv:1603.09355] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    R. Klein and D. Roest, Exorcising the Ostrogradsky ghost in coupled systems, JHEP 07 (2016) 130 [arXiv:1604.01719] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Nicolis, R. Rattazzi and E. Trincherini, The galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].ADSMathSciNetGoogle Scholar
  10. [10]
    C. de Rham and A.J. Tolley, DBI and the galileon reunited, JCAP 05 (2010) 015 [arXiv:1003.5917] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    C. Deffayet, S. Deser and G. Esposito-Farese, Generalized galileons: all scalar models whose curved background extensions maintain second-order field equations and stress-tensors, Phys. Rev. D 80 (2009) 064015 [arXiv:0906.1967] [INSPIRE].ADSGoogle Scholar
  12. [12]
    C. Deffayet, X. Gao, D.A. Steer and G. Zahariade, From k-essence to generalised galileons, Phys. Rev. D 84 (2011) 064039 [arXiv:1103.3260] [INSPIRE].ADSGoogle Scholar
  13. [13]
    C. Deffayet, G. Esposito-Farese and A. Vikman, Covariant galileon, Phys. Rev. D 79 (2009) 084003 [arXiv:0901.1314] [INSPIRE].ADSGoogle Scholar
  14. [14]
    G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  15. [15]
    C. Deffayet, S. Deser and G. Esposito-Farese, Arbitrary p-form galileons, Phys. Rev. D 82 (2010) 061501 [arXiv:1007.5278] [INSPIRE].ADSGoogle Scholar
  16. [16]
    C. Deffayet, S. Mukohyama and V. Sivanesan, On p-form theories with gauge invariant second order field equations, Phys. Rev. D 93 (2016) 085027 [arXiv:1601.01287] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    M. Hull, K. Koyama and G. Tasinato, Covariantized vector galileons, Phys. Rev. D 93 (2016) 064012 [arXiv:1510.07029] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    C. Deffayet and D.A. Steer, A formal introduction to Horndeski and galileon theories and their generalizations, Class. Quant. Grav. 30 (2013) 214006 [arXiv:1307.2450] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Curtright, Generalized gauge fields, Phys. Lett. B 165 (1985) 304 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    P.C. West, The IIA, IIB and eleven-dimensional theories and their common E 11 origin, Nucl. Phys. B 693 (2004) 76 [hep-th/0402140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    P.P. Cook and P.C. West, Charge multiplets and masses for E 11, JHEP 11 (2008) 091 [arXiv:0805.4451] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    F. Riccioni and P.C. West, Dual fields and E 11, Phys. Lett. B 645 (2007) 286 [hep-th/0612001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    E.A. Bergshoeff and F. Riccioni, D-brane Wess-Zumino terms and U-duality, JHEP 11 (2010) 139 [arXiv:1009.4657] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    E.A. Bergshoeff and F. Riccioni, String solitons and T-duality, JHEP 05 (2011) 131 [arXiv:1102.0934] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E.A. Bergshoeff, F. Riccioni and L. Romano, Branes, weights and central charges, JHEP 06 (2013) 019 [arXiv:1303.0221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Chatzistavrakidis, F.F. Gautason, G. Moutsopoulos and M. Zagermann, Effective actions of nongeometric five-branes, Phys. Rev. D 89 (2014) 066004 [arXiv:1309.2653] [INSPIRE].ADSGoogle Scholar
  28. [28]
    A. Chatzistavrakidis and F.F. Gautason, U-dual branes and mixed symmetry tensor fields, Fortsch. Phys. 62 (2014) 743 [arXiv:1404.7635] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    E.A. Bergshoeff, V.A. Penas, F. Riccioni and S. Risoli, Non-geometric fluxes and mixed-symmetry potentials, JHEP 11 (2015) 020 [arXiv:1508.00780] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    P. de Medeiros and C. Hull, Exotic tensor gauge theory and duality, Commun. Math. Phys. 235 (2003) 255 [hep-th/0208155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P. de Medeiros and C. Hull, Geometric second order field equations for general tensor gauge fields, JHEP 05 (2003) 019 [hep-th/0303036] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  32. [32]
    M. Dubois-Violette and M. Henneaux, Tensor fields of mixed Young symmetry type and N complexes, Commun. Math. Phys. 226 (2002) 393 [math/0110088] [INSPIRE].
  33. [33]
    M. Dubois-Violette and M. Henneaux, Generalized cohomology for irreducible tensor fields of mixed Young symmetry type, Lett. Math. Phys. 49 (1999) 245 [math/9907135] [INSPIRE].
  34. [34]
    J.M. Ezquiaga, J. García-Bellido and M. Zumalacárregui, Towards the most general scalar-tensor theories of gravity: a unified approach in the language of differential forms, Phys. Rev. D 94 (2016) 024005 [arXiv:1603.01269] [INSPIRE].
  35. [35]
    N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    C. Aragone and S. Deser, Consistency problems of spin-2 gravity coupling, Nuovo Cim. B 57 (1980) 33 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Hindawi, B.A. Ovrut and D. Waldram, Consistent spin two coupling and quadratic gravitation, Phys. Rev. D 53 (1996) 5583 [hep-th/9509142] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    J. Khoury, J.-L. Lehners and B.A. Ovrut, Supersymmetric galileons, Phys. Rev. D 84 (2011) 043521 [arXiv:1103.0003] [INSPIRE].ADSGoogle Scholar
  39. [39]
    F. Farakos, C. Germani and A. Kehagias, On ghost-free supersymmetric galileons, JHEP 11 (2013) 045 [arXiv:1306.2961] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Padilla, P.M. Saffin and S.-Y. Zhou, Bi-galileon theory I: motivation and formulation, JHEP 12 (2010) 031 [arXiv:1007.5424] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Padilla, P.M. Saffin and S.-Y. Zhou, Multi-galileons, solitons and Derrick’s theorem, Phys. Rev. D 83 (2011) 045009 [arXiv:1008.0745] [INSPIRE].ADSGoogle Scholar
  42. [42]
    N. Deruelle and J. Madore, On the quasilinearity of the Einstein-“Gauss-Bonnet” gravity field equations, gr-qc/0305004 [INSPIRE].
  43. [43]
    K. Van Acoleyen and J. Van Doorsselaere, Galileons from Lovelock actions, Phys. Rev. D 83 (2011) 084025 [arXiv:1102.0487] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Athanasios Chatzistavrakidis
    • 1
    Email author
  • Fech Scen Khoo
    • 2
  • Diederik Roest
    • 1
  • Peter Schupp
    • 2
  1. 1.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Physics and Earth SciencesJacobs University BremenBremenGermany

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