Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 032, 15 pages      arXiv:1201.4247
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

On the Relations between Gravity and BF Theories

Laurent Freidel a and Simone Speziale b
a) Perimeter Institute, 31 Caroline St N, Waterloo ON, N2L 2Y5, Canada
b) Centre de Physique Théorique, CNRS-UMR 7332, Luminy Case 907, 13288 Marseille, France

Received January 23, 2012, in final form May 18, 2012; Published online May 26, 2012

We review, in the light of recent developments, the existing relations between gravity and topological BF theories at the classical level. We include the Plebanski action in both self-dual and non-chiral formulations, their generalizations, and the MacDowell-Mansouri action.

Key words: Plebanski action; MacDowell-Mansouri action; BF gravity; TQFT; modified theories of gravity.

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  1. Alexandrov S., Choice of connection in loop quantum gravity, Phys. Rev. D 65 (2002), 024011, 7 pages, gr-qc/0107071.
  2. Alexandrov S., The Immirzi parameter and fermions with non-minimal coupling, Classical Quantum Gravity 25 (2008), 145012, 4 pages, arXiv:0802.1221.
  3. Alexandrov S., Buffenoir E., Roche P., Plebanski theory and covariant canonical formulation, Classical Quantum Gravity 24 (2007), 2809-2824, gr-qc/0612071.
  4. Alexandrov S., Geiller M., Noui K., Spin foams and canonical quantization, arXiv:1112.1961.
  5. Alexandrov S., Krasnov K., Hamiltonian analysis of non-chiral Plebanski theory and its generalizations, Classical Quantum Gravity 26 (2009), 055005, 10 pages, arXiv:0809.4763.
  6. Alexandrov S., Livine E.R., SU(2) loop quantum gravity seen from covariant theory, Phys. Rev. D 67 (2003), 044009, 15 pages, gr-qc/0209105.
  7. Anishetty R., Vytheeswaran A.S., Gauge invariance in second-class constrained systems, J. Phys. A: Math. Gen. 26 (1993), 5613-5619.
  8. Ashtekar A., New Hamiltonian formulation of general relativity, Phys. Rev. D 36 (1987), 1587-1602.
  9. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  10. Baez J.C., An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Editors H. Gausterer, H. Grosse, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
  11. Baratin A., Oriti D., Group field theory and simplicial gravity path integrals: a model for Holst-Plebanski gravity, Phys. Rev. D 85 (2012), 044003, 15 pages, arXiv:1111.5842.
  12. Barbero G. J.F., Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51 (1995), 5507-5510, gr-qc/9410014.
  13. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2009), 155014, 48 pages, arXiv:0803.3319.
  14. Barros e Sá N., Hamiltonian analysis of general relativity with the Immirzi parameter, Internat. J. Modern Phys. D 10 (2001), 261-272, gr-qc/0006013.
  15. Beke D., Scalar-tensor theories from Λ(φ) Plebanski gravity, arXiv:1111.1139.
  16. Beke D., Palmisano G., Speziale S., Pauli-Fierz mass term in modified Plebanski gravity, J. High Energy Phys. 2012 (2012), no. 3, 069, 28 pages, arXiv:1112.4051.
  17. Benedetti D., Speziale S., Perturbative quantum gravity with the Immirzi parameter, J. High Energy Phys. 2011 (2011), no. 6, 107, 31 pages, arXiv:1104.4028.
  18. Benedetti D., Speziale S., Perturbative running of the Immirzi parameter, arXiv:1111.0884.
  19. Bengtsson I., The cosmological constants, Phys. Lett. B 254 (1991), 55-60.
  20. Bengtsson I., 2-form geometry and the 't Hooft-Plebanski action, Classical Quantum Gravity 12 (1995), 1581-1590, gr-qc/9502010.
  21. Bethke L., Magueijo J., Chirality of tensor perturbations for complex values of the Immirzi parameter, arXiv:1108.0816.
  22. Birmingham D., Blau M., Rakowski M., Thompson G., Topological field theory, Phys. Rep. 209 (1991), 129-340.
  23. Bodendorfer N., Thiemann T., Thurn A., New variables for classical and quantum gravity in all dimensions. I. Hamiltonian analysis, arXiv:1105.3703.
  24. Bodendorfer N., Thiemann T., Thurn A., On the implementation of the canonical quantum simplicity constraint, arXiv:1105.3708.
  25. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincaré 13 (2012), 185-208, arXiv:1103.3961.
  26. Buffenoir E., Henneaux M., Noui K., Roche P., Hamiltonian analysis of Plebanski theory, Classical Quantum Gravity 21 (2004), 5203-5220, gr-qc/0404041.
  27. Capovilla R., Generally covariant gauge theories, Nuclear Phys. B 373 (1992), 233-246.
  28. Capovilla R., Dell J., Jacobson T., Mason L., Self-dual 2-forms and gravity, Classical Quantum Gravity 8 (1991), 41-57.
  29. Capovilla R., Montesinos M., Prieto V.A., Rojas E., BF gravity and the Immirzi parameter, Classical Quantum Gravity 18 (2001), L49-L52, gr-qc/0102073.
  30. Cattaneo A.S., Cotta-Ramusino P., Fröhlich J., Martellini M., Topological BF theories in 3 and 4 dimensions, J. Math. Phys. 36 (1995), 6137-6160, hep-th/9505027.
  31. Cattaneo A.S., Cotta-Ramusino P., Fucito F., Martellini M., Rinaldi M., Tanzini A., Zeni M., Four-dimensional Yang-Mills theory as a deformation of topological BF theory, Comm. Math. Phys. 197 (1998), 571-621, hep-th/9705123.
  32. Cianfrani F., Montani G., Towards loop quantum gravity without the time gauge, Phys. Rev. Lett. 102 (2009), 091301, 4 pages, arXiv:0811.1916.
  33. Clifton T., Bañados M., Skordis C., The parameterized post-Newtonian limit of bimetric theories of gravity, Classical Quantum Gravity 27 (2010), 235020, 31 pages, arXiv:1006.5619.
  34. Damour T., Kogan I.I., Effective Lagrangians and universality classes of nonlinear bigravity, Phys. Rev. D 66 (2002), 104024, 17 pages, hep-th/0206042.
  35. Date G., Kaul R.K., Sengupta S., Topological interpretation of Barbero-Immirzi parameter, Phys. Rev. D 79 (2009), 044008, 7 pages, arXiv:0811.4496.
  36. De Pietri R., Freidel L., so(4) Plebanski action and relativistic spin-foam model, Classical Quantum Gravity 16 (1999), 2187-2196, gr-qc/9804071.
  37. Deruelle N., Sasaki M., Sendouda Y., Yamauchi D., Hamiltonian formulation of f(Riemann) theories of gravity, Progr. Theoret. Phys. 123 (2010), 169-185, arXiv:0908.0679.
  38. Deser S., Teitelboim C., Duality transformations of Abelian and non-Abelian gauge fields, Phys. Rev. D 13 (1976), 1592-1597.
  39. Dona P., Speziale S., Introductory lectures to loop quantum gravity, arXiv:1007.0402.
  40. Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
  41. Dupuis M., Livine E.R., Holomorphic simplicity constraints for 4D spinfoam models, Classical Quantum Gravity 28 (2011), 215022, 32 pages, arXiv:1104.3683.
  42. Durka R., Kowalski-Glikman J., Gravity as a constrained BF theory: Noether charges and Immirzi parameter, Phys. Rev. D 83 (2011), 124011, 6 pages, arXiv:1103.2971.
  43. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  44. Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
  45. Freidel L., Modified gravity without new degrees of freedom, arXiv:0812.3200.
  46. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  47. Freidel L., Krasnov K., Puzio R., BF description of higher-dimensional gravity theories, Adv. Theor. Math. Phys. 3 (1999), 1289-1324, hep-th/9901069.
  48. Freidel L., Louapre D., Diffeomorphisms and spin foam models, Nuclear Phys. B 662 (2003), 279-298, gr-qc/0212001.
  49. Freidel L., Minic D., Takeuchi T., Quantum gravity, torsion, parity violation, and all that, Phys. Rev. D 72 (2005), 104002, 6 pages, hep-th/0507253.
  50. Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, hep-th/0501191.
  51. Geiller M., Lachieze-Rey M., Noui K., A new look at Lorentz-covariant loop quantum gravity, Phys. Rev. D 84 (2011), 044002, 19 pages, arXiv:1105.4194.
  52. Halpern M.B., Field-strength and dual variable formulations of gauge theory, Phys. Rev. D 19 (1979), 517-530.
  53. Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992.
  54. Holst S., Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996), 5966-5969, gr-qc/9511026.
  55. Immirzi G., Real and complex connections for canonical gravity, Classical Quantum Gravity 14 (1997), L177-L181, gr-qc/9612030.
  56. Ishibashi A., Speziale S., Spherically symmetric black holes in minimally modified self-dual gravity, Classical Quantum Gravity 26 (2009), 175005, 37 pages, arXiv:0904.3914.
  57. Krasnov K., Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity, Phys. Rev. Lett. 100 (2008), 081102, 4 pages, arXiv:0711.0090.
  58. Krasnov K., Effective metric Lagrangians from an underlying theory with two propagating degrees of freedom, Phys. Rev. D 81 (2010), 084026, 40 pages, arXiv:0911.4903.
  59. Krasnov K., Renormalizable non-metric quantum gravity?, hep-th/0611182.
  60. Krasnov K., Shtanov Y., Cosmological perturbations in a family of deformations of general relativity, J. Cosmol. Astropart. Phys. 2010 (2010), no. 6, 006, 42 pages, arXiv:1002.1210.
  61. Krasnov K., Shtanov Y., Halos of modified gravity, Internat. J. Modern Phys. D 17 (2008), 2555-2562, arXiv:0805.2668.
  62. Lisi A.G., An exceptionally simple theory of everything, arXiv:0711.0770.
  63. Lisi A.G., Smolin L., Speziale S., Unification of gravity, gauge fields and Higgs bosons, J. Phys. A: Math. Theor. 43 (2010), 445401, 10 pages, arXiv:1004.4866.
  64. Liu L., Montesinos M., Perez A., Topological limit of gravity admitting an SU(2) connection formulation, Phys. Rev. D 81 (2010), 064033, 9 pages, arXiv:0906.4524.
  65. Livine E.R., Speziale S., Solving the simplicity constraints for spinfoam quantum gravity, Europhys. Lett. 81 (2008), 50004, 6 pages, arXiv:0708.1915.
  66. MacDowell S.W., Mansouri F., Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977), 739-742.
  67. Mercuri S., Fermions in the Ashtekar-Barbero connection formalism for arbitrary values of the Immirzi parameter, Phys. Rev. D 73 (2006), 084016, 14 pages, gr-qc/0601013.
  68. Mielke E.W., Spontaneously broken topological SL(5,R) gauge theory with standard gravity emerging, Phys. Rev. D 83 (2011), 044004, 9 pages.
  69. Mitra P., Rajaraman R., Gauge-invariant reformulation of theories with second-class constraints, Ann. Physics 203 (1990), 157-172.
  70. Montesinos M., Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories, Classical Quantum Gravity 23 (2006), 2267-2278, gr-qc/0603076.
  71. Montesinos M., Velázquez M., BF gravity with Immirzi parameter and cosmological constant, Phys. Rev. D 81 (2010), 044033, 4 pages, arXiv:1002.3836.
  72. Peldán P., Actions for gravity, with generalizations: a review, Classical Quantum Gravity 11 (1994), 1087-1132, gr-qc/9305011.
  73. Percacci R., Gravity from a particle physicists' perspective, PoS Proc. Sci. (2009), PoS(ISFTG2009), 011, 30 pages, arXiv:0910.5167.
  74. Perez A., Spin foam quantization of SO(4) Pleba\'nski's action, Adv. Theor. Math. Phys. 5 (2001), 947-968, gr-qc/0203058.
  75. Perez A., The spin foam approach to quantum gravity, Living Rev. Relativ., to appear, arXiv:1205.2019.
  76. Perez A., Rovelli C., Physical effects of the Immirzi parameter in loop quantum gravity, Phys. Rev. D 73 (2006), 044013, 3 pages, gr-qc/0505081.
  77. Plebanski J.F., On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977), 2511-2520.
  78. Randono A., de Sitter spaces: topological ramifications of gravity as a gauge theory, Classical Quantum Gravity 27 (2010), 105008, 18 pages, arXiv:0909.5435.
  79. Reisenberger M.P., A left-handed simplicial action for Euclidean general relativity, Classical Quantum Gravity 14 (1997), 1753-1770, gr-qc/9609002.
  80. Reisenberger M.P., Classical Euclidean general relativity from "left-handed area = right-handed area", Classical Quantum Gravity 16 (1999), 1357-1371, gr-qc/9804061.
  81. Reisenberger M.P., New constraints for canonical general relativity, Nuclear Phys. B 457 (1995), 643-687, gr-qc/9505044.
  82. Rivasseau V., Towards renormalizing group field theory, PoS Proc. Sci. (2010), PoS(CNCFG2010), 004, 21 pages, arXiv:1103.1900.
  83. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  84. Rovelli C., Speziale S., On the expansion of a quantum field theory around a topological sector, Gen. Relativity Gravitation 39 (2007), 167-178, gr-qc/0508106.
  85. Smolin L., Plebanski action extended to a unification of gravity and Yang-Mills theory, Phys. Rev. D 80 (2009), 124017, 6 pages, arXiv:0712.0977.
  86. Smolin L., Speziale S., Note on the Plebanski action with the cosmological constant and an Immirzi parameter, Phys. Rev. D 81 (2010), 024032, 6 pages, arXiv:0908.3388.
  87. Speziale S., Bimetric theory of gravity from the nonchiral Plebanski action, Phys. Rev. D 82 (2010), 064003, 17 pages, arXiv:1003.4701.
  88. Stelle K.S., Classical gravity with higher derivatives, Gen. Relativity Gravitation 9 (1978), 353-371.
  89. Stelle K.S., West P.C., de Sitter gauge invariance and the geometry of the Einstein-Cartan theory, J. Phys. A: Math. Gen. 12 (1979), L205-L210.
  90. 't Hooft G., A chiral alternative to the vierbein field in general relativity, Nuclear Phys. B 357 (1991), 211-221.
  91. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  92. Townsend P.K., Small-scale structure of spacetime as the origin of the gravitational constant, Phys. Rev. D 15 (1977), 2795-2801.
  93. Tseytlin A.A., Poincaré and de Sitter gauge theories of gravity with propagating torsion, Phys. Rev. D 26 (1982), 3327-3341.
  94. Urbantke H., On integrability properties of SU(2) Yang-Mills fields. I. Infinitesimal part, J. Math. Phys. 25 (1984), 2321-2324.
  95. Wieland W.M., Complex Ashtekar variables and reality conditions for Holst's action, Ann. Henri Poincaré 13 (2012), 425-448, arXiv:1012.1738.
  96. Wilczek F., Riemann-Einstein structure from volume and gauge symmetry, Phys. Rev. Lett. 80 (1998), 4851-4854, hep-th/9801184.
  97. Wise D.K., MacDowell-Mansouri gravity and Cartan geometry, Classical Quantum Gravity 27 (2010), 155010, 26 pages, gr-qc/0611154.
  98. Witten E., 2+1-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
  99. Zapata J.A., Topological lattice gravity using self-dual variables, Classical Quantum Gravity 13 (1996), 2617-2634, gr-qc/9603030.

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