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.
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References
Ashtekar, A., Lewandowski, J.: Quantum theory of geometry. 1: Area operators. Class. Quantum Gravity 14, A55 (1997)
Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53 (2004)
Ashtekar, A., Baez, J., Corichi, A., Krasnov, K.: Quantum geometry and black hole entropy. Phys. Rev. Lett. 80, 904 (1998)
Banados, M., Ferreira, P.G.: Eddington’s theory of gravity and its progeny. Phys. Rev. Lett. 105, 011101 (2010)
Blagojević, M., Hehl, F.W. (eds.): Gauge Theories of Gravitation, a Reader with Commentaries. Imperial College Press, London (2013)
Born, M., Infeld, L.: Foundations of the new field theory. Nature 132, 1004 (1933)
Chagoya, J., Sabido, M.: Topological M-theory, self dual gravity and the Immirzi parameter. Class. Quantum Gravity 35, 165002 (2018)
Chamseddine, A.H.: Massive supergravity from spontaneously breaking orthosymplectic gauge symmetry. Ann. Phys. 113, 219 (1978)
Comelli, D., Dolgov, A.: Determinant-gravity: cosmological implications. JHEP 11, 062 (2004)
Deser, S., Gibbons, G.: Born–Infeld–Einstein actions? Class. Quantum Gravity 15, L.35 (1998)
Eddington, A.S.: The Mathematical Theory of Relativity. Cambridge University Press, Cambridge (1924)
Feigenbaum, J.A.: Born-regulated gravity in four dimensions. Phys. Rev. D 58, 124023 (1998)
Gambini, R., Obregon, O., Pullin, J.: Yang–Mills analogs of the Immirzi ambiguity. Phys. Rev. D 59, 047505 (1999)
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)
García-Compeán, H., Obregón, O., Ramírez, C.: Gravitational duality in MacDowell–Mansouri gauge theory. Phys. Rev. D 58, 104012 (1998)
Gotzes, S., Hirshfeld, A.C.: A geometric formulation of the \(SO(3,2)\) theory of gravity. Ann. Phys. 203, 410 (1990)
Gullu, I., Sisman, T.C., Tekin, B.: Born–Infeld gravity with a massless graviton in four dimensions. Phys. Rev. D 91(4), 044007 (2015)
Immirzi, G.: Real and complex connections for canonical gravity. Class. Quantum Gravity 14, L177 (1997)
Jacobson, T., Smolin, L.: Covariant action for Ashtekar’s form of canonical gravity. Class. Quantum Gravity 5, 583 (1988)
Jimenez, J.B., Heisengerg, L., Olmo, G., Rubiera-Garcia, D.: Born–Infeld modifications of gravity. Phys. Rep. 727, 1 (2017)
MacDowell, S.W., Mansouri, F.: Unified geometric theory of gravity and supergravity. Phys. Rev. Lett. 38, 739 (1997)
Mansouri, F.: Superunified theories based on the geometry of local (super-) gauge invariance. Phys. Rev. D 16, 2456 (1977)
Mercuri, S.: Fermions in Ashtekar–Barbero connections formalism for arbitrary values of the Immirzi parameter. Phys. Rev. D 73, 084016 (2006)
Mercuri, S., Randono, A.: The Immirzi parameter as an instanton angle. Class. Quantum Gravity 28, 025001 (2011)
Nieto, J.A.: Born–Infeld gravity in any dimension. Phys. Rev. D 70, 044042 (2004)
Nieto, J.A., Obregón, O., Socorro, J.: Gauge theory of supergravity based only on a selfdual spin connection. Phys. Rev. Lett. 76, 3482 (1996)
Nieto, J.A., Obregón, O., Socorro, J.: The gauge theory of the de Sitter group and Ashtekar formulation. Phys. Rev. D 50, R3583 (1994)
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)
Obregón, O.: Non-abelian Born–Infeld theory without the square root. Mod. Phys. Lett. A 21, 1249 (2006)
Obregón, O., Ortega-Cruz, M., Sabido, M.: Immirzi parameter and \(\theta \) ambiguity in de Sitter MacDowell–Mansouri supergravity. Phys. Rev. D 85, 124061 (2012)
Ortín, T.: Gravity and Strings. Cambridge Monographs of Mathematics and Physics. Cambridge University Press, Cambridge (2004)
Polchinski, J.: String Theory, vols. 1, 2. An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1998)
Rovelli, C., Smolin, L.: Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593 (1995)
Rovelli, C., Thiemann, T.: The Immirzi parameter in quantum general relativity. Phys. Rev. D 57, 1009 (1998)
Samuel, J.: A Lagrangian basis for Ashtekar’s formulation of canonical gravity. Pramana J. Phys. 28, L429 (1987)
Schrodinger, E.: Contribution to Born’s new theory of the electromagnetic field. Proc. R. Soc. A 150, 465 (1935)
Tseytlin, A.A.: On non-abelian generalization of Born–Infeld action in string theory. Nucl. Phys. B 501, 41–52 (1997)
Vollick, D.N.: Palatini approach to Born–Infeld–Einstein theory and a geometric description of electrodynamics. Phys. Rev. D 69, 064030 (2004)
West, P.C.: A geometric gravity Lagrangian. Phys. Lett. 76B, 569 (1978)
Wise, Derek K.: MacDowell–Mansouri gravity and Cartan geometry. Class. Quantum Gravity 27, 155010 (2010)
Wohlfarth, M.N.R.: Gravity a la Born–Infeld. Class. Quantum Gravity 21, 1927 (2004)
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.
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Dedicated to the memory of Professor Waldyr A. Rodrigues Jr..
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López, J.L., Obregón, O. & Ortega-Cruz, M. Born–Infeld Gravity from the MacDowell–Mansouri Action and Its Associated \({\varvec{\beta }}\)-Term. Adv. Appl. Clifford Algebras 29, 24 (2019). https://doi.org/10.1007/s00006-019-0940-9
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DOI: https://doi.org/10.1007/s00006-019-0940-9