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

SIGMA 5 (2009), 080, 18 pages      arXiv:0904.1738
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions

Derek K. Wise
Department of Mathematics, University of California, Davis, CA 95616, USA

Received April 10, 2009, in final form July 19, 2009; Published online August 01, 2009

Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.

Key words: Cartan geometry; symmetric spaces; general relativity; Chern-Simons theory; topologically massive gravity; MacDowell-Mansouri gravity.

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  1. Achúcarro A., Townsend P.K., A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986), 89-92.
  2. Aldinger R.R., Geometrical SO(4,1) gauge theory as a basis of extended relativistic objects for hadrons, J. Phys. A: Math. Gen. 23 (1990), 1885-1907.
  3. Alekseevsky D.V., Michor P.W., Differential geometry of Cartan connections, Publ. Math. Debrecen 47 (1995), 349-375, math.DG/9412232.
  4. Carlip S., Quantum gravity in 2+1 dimensions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1998.
  5. Carlip S., Geometric structures and loop variables in (2+1)-dimensional gravity, in Knots and Quantum Gravity (Riverside, CA, 1993, Editor J. Baez, Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford Univ. Press, New York, 1994 97-111, gr-qc/9309020.
  6. Carlip S., The constraint algebra of topologically massive AdS gravity, J. High Energy Phys. 2008 (2008), no. 10, 078, 19 pages, arXiv:0807.4152.
  7. Carlip S., Deser S., Waldron A., Wise D.K., Cosmological topologically massive gravitons and photons, Classical Quantum Gravity 26 (2009), 075008, 24 pages, arXiv:0803.3998.
    Carlip S., Deser S., Waldron A., Wise D.K., Topologically massive AdS gravity, Phys. Lett. B 666 (2008), 272-276, arXiv:0807.0486.
  8. Cartan É., Les espaces à connexion conforme, Ann. Soc. Pol. Math.  2 (1923), 171-221.
  9. Cartan É., Les groupes d'holonomie des espaces généralisés, Acta Math. 48 (1926), 1-42.
  10. Cartan É., La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés, Exposés de Géométrie, No. 5, Hermann, Paris, 1935.
  11. Cartan É., Les groupes prejectifs qui ne laissent invariante aucune multiplicité plane, Bul. Soc. Math. France 41 (1913), 53-96.
    Cartan É., The theory of spinors, Dover, New York, 1981 (a reprint of the 1966 English translation).
  12. Cartan É., On a generalization of the notion of Riemann curvature and spaces with torsion, in Cosmology and Gravitation (Bologna, 1979), Editors P.G. Bergmann and V. De Sabbata, NATO Adv. Study Inst. Ser. B: Physics, Vol. 58, Plenum Press, New York - London, 1980, 489-491 (Translation by G. D. Kerlick from French original article: Sur une generalisation de la notion de courbure de Riemann et les espaces a torsion, Comp. Rend. Acad. Sci. Paris 174 (1922), 593-595).
  13. Cartan É., Einstein A., Élie Cartan and Albert Einstein: Letters on absolute parallelism, 1929-1932, Editor R. Devever, Princeton University Press, Princeton, N.J.; Academie Royale de Belgique, Brussels, 1979.
  14. Chern S.-S., Chevalley C., Obituary: Élie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 217-250.
  15. Deser S., Jackiw R., Templeton S., Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982), 975-978.
    Deser S., Jackiw R., Templeton S., Topologically massive gauge theories, Ann. Physics 140 (1982), 372-411.
  16. Deser S., A note on first-order formalism and odd-derivative actions, Classical Quantum Gravity 23 (2006), 5773-5776, gr-qc/0606006.
  17. Ehresmann C., Sur les espaces localement homogenes, L'ens. Math. 35 (1936), 317-333.
  18. Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, hep-th/0501191.
  19. Gibbons G.W., Gielen S., Deformed general relativity and torsion, Classical Quantum Gravity 26 (2009), 135005, 18 pages, arXiv:0902.2001.
  20. Goldman W.M., Topological components of spaces of representations, Invent. Math. 93 (1988), 557-607.
  21. Goldman W.M., Geometric structures and varieties of representations, in The Geometry of Group Representations (Boulder, CO, 1987), Contemp. Math., Vol. 74, Amer. Math. Soc., Providence, RI, 1988, 169-198.
  22. Hehl F.W., McCrea J.D., Mielke E.W., Ne'eman Y., Metric affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rept. 258 (1995), 1-171, gr-qc/9402012.
  23. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
  24. Ikeda N., Fukuyama T., Fermions in (anti) de Sitter gravity in four dimensions, arXiv:0904.1936.
  25. Kobayashi S., On connections of Cartan, Canad. J. Math. 8 (1956), 145-156.
  26. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, John Wiley & Sons, Inc., New York, 1996.
  27. Li W., Song W., Strominger A., Chiral gravity in three dimensions, J. High Energy Phys. 2008 (2008), no. 4, 082, 15 pages, arXiv:0801.4566.
  28. MacDowell S.W., Mansouri F., Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977), 739-742, Erratum, Phys. Rev. Lett. 38 (1977), 1376.
  29. Mielke E.W., Obukhov Y.N., Hehl F.W., The Yang-Mills configurations from 3-D Riemann-Cartan geometry, Phys. Lett. A 192 (1994), 153-162, gr-qc/9407031.
  30. Misner C.W., Thorne K.S., Wheeler J.A., Gravitation, W.H. Freeman and Co., San Francisco, Calif., 1973.
  31. Nieto J.A., Socorro J., Obregon O., Gauge theory of the de Sitter group and the Ashtekar formulation, Phys. Rev. D 50 (1994), R3583-R3586, gr-qc/9402029.
    Nieto J.A., Socorro J., Obregon O., Gauge theory of supergravity based only on a selfdual spin connection, Phys. Rev. Lett. 76 (1996), 3482-3485.
  32. Randono A., Generalizing the Kodama state II: Properties and physical interpretation, gr-qc/0611074.
  33. Ruh E., Cartan connections, in Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Vol. 54, Part 3, Amer. Math. Soc., Providence, RI, 1993.
  34. Sharpe R.W., Differential geometry: Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.
  35. Thurston W.P., Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997.
  36. Thurston W.P., The geometry and topology of three-manifolds, available at
  37. Trautman A., Einstein-Cartan theory, Encyclopedia of Mathematical Physics, Editors J.-P. Francoise, G.L. Naber and S.T. Tsou, Oxford, Elsevier, 2006, Vol. 2, 189-195, gr-qc/0606062.
  38. Wise D.K., MacDowell-Mansouri gravity and Cartan geometry, gr-qc/0611154.
  39. Witten E., (2+1)-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
  40. Zardecki A., Gravity as a gauge theory with Cartan connection, J. Math. Phys. 29 (1988), 1661-1666.

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