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Born–Infeld Gravity from the MacDowell–Mansouri Action and Its Associated \({\varvec{\beta }}\)-Term

  • J. L. LópezEmail author
  • O. Obregón
  • M. Ortega-Cruz
Article
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Part of the following topical collections:
  1. Homage to Prof. W.A. Rodrigues Jr

Abstract

In this work a generalization of a Born–Infeld theory of gravity with a topological \(\beta \)-term is proposed. These type of Born–Infeld actions were found from the theory introduced by MacDowell and Mansouri. This theory known as MacDowell–Mansouri (MM) gravity was one of the first attempts to construct a gauge theory of gravitation, and within this framework it was introduced in the action a topological \(\beta \)-term relevant for quantization purposes in an analogous way as in Yang–Mills theory. By the use of the self-dual and antiself-dual actions of MM gravity, we further define a Born–Infeld gravity generalization corresponding to MM gravity with the \(\beta \)-term.

Keywords

Born–Infeld gravity MacDowell–Mansouri gravity Self-dual gravity 

Notes

Acknowledgements

O. Obregón thanks CONACYT Project 257919, UG Proyect CIIC 130/2018 and Prodep Projects. J. L. López acknowledge CONACYT, UG and PRODEP Grant 511-6/18-8876.

References

  1. 1.
    Ashtekar, A., Lewandowski, J.: Quantum theory of geometry. 1: Area operators. Class. Quantum Gravity 14, A55 (1997)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53 (2004)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ashtekar, A., Baez, J., Corichi, A., Krasnov, K.: Quantum geometry and black hole entropy. Phys. Rev. Lett. 80, 904 (1998)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Banados, M., Ferreira, P.G.: Eddington’s theory of gravity and its progeny. Phys. Rev. Lett. 105, 011101 (2010)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Blagojević, M., Hehl, F.W. (eds.): Gauge Theories of Gravitation, a Reader with Commentaries. Imperial College Press, London (2013)zbMATHGoogle Scholar
  6. 6.
    Born, M., Infeld, L.: Foundations of the new field theory. Nature 132, 1004 (1933)ADSCrossRefGoogle Scholar
  7. 7.
    Chagoya, J., Sabido, M.: Topological M-theory, self dual gravity and the Immirzi parameter. Class. Quantum Gravity 35, 165002 (2018)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chamseddine, A.H.: Massive supergravity from spontaneously breaking orthosymplectic gauge symmetry. Ann. Phys. 113, 219 (1978)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Comelli, D., Dolgov, A.: Determinant-gravity: cosmological implications. JHEP 11, 062 (2004)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Deser, S., Gibbons, G.: Born–Infeld–Einstein actions? Class. Quantum Gravity 15, L.35 (1998)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Eddington, A.S.: The Mathematical Theory of Relativity. Cambridge University Press, Cambridge (1924)zbMATHGoogle Scholar
  12. 12.
    Feigenbaum, J.A.: Born-regulated gravity in four dimensions. Phys. Rev. D 58, 124023 (1998)ADSCrossRefGoogle Scholar
  13. 13.
    Gambini, R., Obregon, O., Pullin, J.: Yang–Mills analogs of the Immirzi ambiguity. Phys. Rev. D 59, 047505 (1999)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    García-Compeán, H., Obregón, O., Plebanski, J.F., Ramírez, C.: Towards a gravitational analog to \(S\) duality in non-abelian gauge theories. Phys. Rev. D. 57, 7501 (1998)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    García-Compeán, H., Obregón, O., Ramírez, C.: Gravitational duality in MacDowell–Mansouri gauge theory. Phys. Rev. D 58, 104012 (1998)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gotzes, S., Hirshfeld, A.C.: A geometric formulation of the \(SO(3,2)\) theory of gravity. Ann. Phys. 203, 410 (1990)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gullu, I., Sisman, T.C., Tekin, B.: Born–Infeld gravity with a massless graviton in four dimensions. Phys. Rev. D 91(4), 044007 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Immirzi, G.: Real and complex connections for canonical gravity. Class. Quantum Gravity 14, L177 (1997)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jacobson, T., Smolin, L.: Covariant action for Ashtekar’s form of canonical gravity. Class. Quantum Gravity 5, 583 (1988)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Jimenez, J.B., Heisengerg, L., Olmo, G., Rubiera-Garcia, D.: Born–Infeld modifications of gravity. Phys. Rep. 727, 1 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    MacDowell, S.W., Mansouri, F.: Unified geometric theory of gravity and supergravity. Phys. Rev. Lett. 38, 739 (1997)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Mansouri, F.: Superunified theories based on the geometry of local (super-) gauge invariance. Phys. Rev. D 16, 2456 (1977)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Mercuri, S.: Fermions in Ashtekar–Barbero connections formalism for arbitrary values of the Immirzi parameter. Phys. Rev. D 73, 084016 (2006)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mercuri, S., Randono, A.: The Immirzi parameter as an instanton angle. Class. Quantum Gravity 28, 025001 (2011)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Nieto, J.A.: Born–Infeld gravity in any dimension. Phys. Rev. D 70, 044042 (2004)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Nieto, J.A., Obregón, O., Socorro, J.: Gauge theory of supergravity based only on a selfdual spin connection. Phys. Rev. Lett. 76, 3482 (1996)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Nieto, J.A., Obregón, O., Socorro, J.: The gauge theory of the de Sitter group and Ashtekar formulation. Phys. Rev. D 50, R3583 (1994)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Nieto, J.A., Socorro, J.: Selfdual gravity and selfdual Yang–Mills theory in the context of MacDowell–Mansouri formalism. Phys. Rev. D 59, 041501 (1999)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Obregón, O.: Non-abelian Born–Infeld theory without the square root. Mod. Phys. Lett. A 21, 1249 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Obregón, O., Ortega-Cruz, M., Sabido, M.: Immirzi parameter and \(\theta \) ambiguity in de Sitter MacDowell–Mansouri supergravity. Phys. Rev. D 85, 124061 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    Ortín, T.: Gravity and Strings. Cambridge Monographs of Mathematics and Physics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  32. 32.
    Polchinski, J.: String Theory, vols. 1, 2. An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1998)Google Scholar
  33. 33.
    Rovelli, C., Smolin, L.: Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593 (1995)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Rovelli, C., Thiemann, T.: The Immirzi parameter in quantum general relativity. Phys. Rev. D 57, 1009 (1998)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Samuel, J.: A Lagrangian basis for Ashtekar’s formulation of canonical gravity. Pramana J. Phys. 28, L429 (1987)ADSCrossRefGoogle Scholar
  36. 36.
    Schrodinger, E.: Contribution to Born’s new theory of the electromagnetic field. Proc. R. Soc. A 150, 465 (1935)ADSzbMATHGoogle Scholar
  37. 37.
    Tseytlin, A.A.: On non-abelian generalization of Born–Infeld action in string theory. Nucl. Phys. B 501, 41–52 (1997)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Vollick, D.N.: Palatini approach to Born–Infeld–Einstein theory and a geometric description of electrodynamics. Phys. Rev. D 69, 064030 (2004)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    West, P.C.: A geometric gravity Lagrangian. Phys. Lett. 76B, 569 (1978)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Wise, Derek K.: MacDowell–Mansouri gravity and Cartan geometry. Class. Quantum Gravity 27, 155010 (2010)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Wohlfarth, M.N.R.: Gravity a la Born–Infeld. Class. Quantum Gravity 21, 1927 (2004)ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de Física, División de Ciencias e Ingenierías Campus LeónUniversidad de GuanajuatoLeónMexico

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