Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Off-shell unimodular N = 1, d = 4 supergravity

  • 34 Accesses


We formulate a unimodular N = 1, d = 4 supergravity theory off shell. We see that the infinitesimal Grassmann parameters defining the unimodular supergravity trans- formations are constrained and show that the conmutator of two infinitesinal unimodular supergravity transformations closes on transverse diffeomorphisms, Lorentz transforma- tions and unimodular supergravity transformations. Along the way, we also show that the linearized theory is a supersymmetric theory of gravitons and gravitinos. We see that de Sitter and anti-de Sitter spacetimes are non-supersymmetric vacua of our unimodular supergravity theory.

A preprint version of the article is available at ArXiv.


  1. [1]

    J.J. van der Bij, H. van Dam and Y.J. Ng, The Exchange of Massless Spin Two Particles, PhysicaA 116 (1982) 307.

  2. [2]

    W. Buchmüller and N. Dragon, Einstein Gravity From Restricted Coordinate Invariance, Phys. Lett.B 207 (1988) 292 [INSPIRE].

  3. [3]

    W. Buchmüller and N. Dragon, Gauge Fixing and the Cosmological Constant, Phys. Lett.B 223 (1989) 313 [INSPIRE].

  4. [4]

    M. Henneaux and C. Teitelboim, The Cosmological Constant and General Covariance, Phys. Lett.B 222 (1989) 195 [INSPIRE].

  5. [5]

    G.F.R. Ellis, H. van Elst, J. Murugan and J.-P. Uzan, On the Trace-Free Einstein Equations as a Viable Alternative to General Relativity, Class. Quant. Grav.28 (2011) 225007 [arXiv:1008.1196] [INSPIRE].

  6. [6]

    G.F.R. Ellis, The Trace-Free Einstein Equations and inflation, Gen. Rel. Grav.46 (2014) 1619 [arXiv:1306.3021] [INSPIRE].

  7. [7]

    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys.61 (1989) 1 [INSPIRE].

  8. [8]

    L. Smolin, The Quantization of unimodular gravity and the cosmological constant problems, Phys. Rev.D 80 (2009) 084003 [arXiv:0904.4841] [INSPIRE].

  9. [9]

    E. Alvarez, Can one tell Einstein’s unimodular theory from Einstein’s general relativity?, JHEP03 (2005) 002 [hep-th/0501146] [INSPIRE].

  10. [10]

    J. Kluson, Canonical Analysis of Unimodular Gravity, Phys. Rev.D 91 (2015) 064058 [arXiv:1409.8014] [INSPIRE].

  11. [11]

    I.D. Saltas, UV structure of quantum unimodular gravity, Phys. Rev.D 90 (2014) 124052 [arXiv:1410.6163] [INSPIRE].

  12. [12]

    A. Eichhorn, The Renormalization Group flow of unimodular f(R) gravity, JHEP04 (2015) 096 [arXiv:1501.05848] [INSPIRE].

  13. [13]

    E. Álvarez, S. González-Martín, M. Herrero-Valea and C.P. Martín, Quantum Corrections to Unimodular Gravity, JHEP08 (2015) 078 [arXiv:1505.01995] [INSPIRE].

  14. [14]

    R. Bufalo, M. Oksanen and A. Tureanu, How unimodular gravity theories differ from general relativity at quantum level, Eur. Phys. J.C 75 (2015) 477 [arXiv:1505.04978] [INSPIRE].

  15. [15]

    E. Alvarez, S. Gonzalez-Martin and C.P. Martin, Unimodular Trees versus Einstein Trees, Eur. Phys. J.C 76 (2016) 554 [arXiv:1605.02667] [INSPIRE].

  16. [16]

    C.P. Martin, Unimodular Gravity and the lepton anomalous magnetic moment at one-loop, JCAP07 (2017) 019 [arXiv:1704.01818] [INSPIRE].

  17. [17]

    S. Gonzalez-Martin and C.P. Martin, Do the gravitational corrections to the β-functions of the quartic and Yukawa couplings have an intrinsic physical meaning?, Phys. Lett.B 773 (2017) 585 [arXiv:1707.06667] [INSPIRE].

  18. [18]

    R. de León Ardón, N. Ohta and R. Percacci, Path integral of unimodular gravity, Phys. Rev.D 97 (2018) 026007 [arXiv:1710.02457] [INSPIRE].

  19. [19]

    R. Percacci, Unimodular quantum gravity and the cosmological constant, Found. Phys.48 (2018) 1364 [arXiv:1712.09903] [INSPIRE].

  20. [20]

    S. González-Martín and C.P. Martin, Unimodular Gravity and General Relativity UV divergent contributions to the scattering of massive scalar particles, JCAP01 (2018) 028 [arXiv:1711.08009] [INSPIRE].

  21. [21]

    M. Herrero-Valea, What do gravitons say about (unimodular) gravity?, JHEP12 (2018) 106 [arXiv:1806.01869] [INSPIRE].

  22. [22]

    G.P. De Brito, A. Eichhorn and A.D. Pereira, A link that matters: Towards phenomenological tests of unimodular asymptotic safety, JHEP09 (2019) 100 [arXiv:1907.11173] [INSPIRE].

  23. [23]

    D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Progress Toward a Theory of Supergravity, Phys. Rev.D 13 (1976) 3214 [INSPIRE].

  24. [24]

    S. Deser and B. Zumino, Consistent Supergravity, Phys. Lett.62B (1976) 335 [INSPIRE].

  25. [25]

    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press (2012) [INSPIRE].

  26. [26]

    S. Ferrara and P. van Nieuwenhuizen, The Auxiliary Fields of Supergravity, Phys. Lett.74B (1978) 333 [INSPIRE].

  27. [27]

    K.S. Stelle and P.C. West, Minimal Auxiliary Fields for Supergravity, Phys. Lett.74B (1978) 330 [INSPIRE].

  28. [28]

    H. Nishino and S. Rajpoot, Unimodular supergravity, Phys. Lett.B 528 (2002) 259 [hep-th/0107202] [INSPIRE].

  29. [29]

    S.J. Gates Jr., W.D. Linch III, J. Phillips and L. Rana, The Fundamental supersymmetry challenge remains, Grav. Cosmol.8 (2002) 96 [hep-th/0109109] [INSPIRE].

  30. [30]

    W. Siegel and S.J. Gates Jr., Superfield Supergravity, Nucl. Phys.B 147 (1979) 77 [INSPIRE].

  31. [31]

    S. Nagy, A. Padilla and I. Zavala, The Super-Stückelberg procedure and dS in Pure Supergravity, arXiv:1910.14349 [INSPIRE].

  32. [32]

    E. Alvarez, D. Blas, J. Garriga and E. Verdaguer, Transverse Fierz-Pauli symmetry, Nucl. Phys.B 756 (2006) 148 [hep-th/0606019] [INSPIRE].

  33. [33]

    W. Rarita and J. Schwinger, On a theory of particles with half integral spin, Phys. Rev.60 (1941) 61 [INSPIRE].

  34. [34]

    D. Blas, Transverse Symmetry and Spin-3/2 Fields, Class. Quant. Grav.25 (2008) 154009 [arXiv:0803.4497] [INSPIRE].

  35. [35]

    P. Van Nieuwenhuizen, Supergravity, Phys. Rept.68 (1981) 189 [INSPIRE].

  36. [36]

    P.C. West, Introduction To Supersymmetry And Supergravity, Singapore, Singapore, World Scientific (1986) [INSPIRE].

  37. [37]

    E. Álvarez and S. González-Martín, First Order formulation of Unimodular Gravity, Phys. Rev.D 92 (2015) 024036 [arXiv:1506.07410] [INSPIRE].

  38. [38]

    C. Bohle, Killing spinors on Lorentzian manifolds, J. Geom. Phys.45 (2003) 285 [INSPIRE].

  39. [39]

    T. Ortin, Gravity and Strings, Cambridge University Press (2015) [INSPIRE].

  40. [40]

    P.K. Townsend, Cosmological Constant in Supergravity, Phys. Rev.D 15 (1977) 2802 [INSPIRE].

  41. [41]

    M. Kaku and P.K. Townsend, Poincaré supergravity as broken superconformal gravity, Phys. Lett.76B (1978) 54 [INSPIRE].

  42. [42]

    S. Ferrara and P. van Nieuwenhuizen, Tensor Calculus for Supergravity, Phys. Lett.76B (1978) 404 [INSPIRE].

  43. [43]

    W. Siegel, Solution to Constraints in Wess-Zumino Supergravity Formalism, Nucl. Phys.B 142 (1978) 301 [INSPIRE].

  44. [44]

    J. Anero, Carmelo P. Martin and R. Santos-Garcia, A note on unimodular N = 1; d = 4 AdS supergravity, to appear.

  45. [45]

    E.A. Bergshoeff, D.Z. Freedman, R. Kallosh and A. Van Proeyen, Pure de Sitter Supergravity, Phys. Rev.D 92 (2015) 085040 [Erratum ibid.D 93 (2016) 069901] [arXiv:1507.08264] [INSPIRE].

  46. [46]

    F. Hasegawa and Y. Yamada, Component action of nilpotent multiplet coupled to matter in 4 dimensional \( \mathcal{N} \) = 1 supergravity, JHEP10 (2015) 106 [arXiv:1507.08619] [INSPIRE].

  47. [47]

    E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti, Properties of Nilpotent Supergravity, JHEP09 (2015) 217 [arXiv:1507.07842] [INSPIRE].

  48. [48]

    S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Complex three-form supergravity and membranes, JHEP12 (2017) 005 [arXiv:1710.00535] [INSPIRE].

  49. [49]

    F. Farakos, S. Lanza, L. Martucci and D. Sorokin, Three-forms, Supersymmetry and String Compactifications, Phys. Part. Nucl.49 (2018) 823 [arXiv:1712.09366] [INSPIRE].

  50. [50]

    I. Bandos, F. Farakos, S. Lanza, L. Martucci and D. Sorokin, Three-forms, dualities and membranes in four-dimensional supergravity, JHEP07 (2018) 028 [arXiv:1803.01405] [INSPIRE].

Download references

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

Author information

Correspondence to Carmelo P. Martin.

Additional information

ArXiv ePrint: 1911.04160

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anero, J., Martin, C.P. & Santos-Garcia, R. Off-shell unimodular N = 1, d = 4 supergravity. J. High Energ. Phys. 2020, 145 (2020). https://doi.org/10.1007/JHEP01(2020)145

Download citation


  • Models of Quantum Gravity
  • Supergravity Models