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

SIGMA 5 (2009), 080, 18 pages      arXiv:0904.1738      https://doi.org/10.3842/SIGMA.2009.080
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

Abstract
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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