This thread needs updating. Quite a lot of research has come out in the wake of what
John Baez and Garrett Lisi were discussing here. I will try to catch it up a little. Others' help woud be very welcome.
In fact, JB and Garrett were talking about some work by Derek Wise and as it happens Derek published a paper just last month (August 2009) which continues the line of research they were discussing.
http://arxiv.org/abs/0904.1738
Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
Derek K. Wise
Article prepared for special journal issue dedicated to Elie Cartan
SIGMA 5 (2009), 080, 18 pages
(Submitted on 10 Apr 2009, revised for publication 3 Aug 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'."
Derek will be giving a talk next week in Corfu, on Saturday 19 September.
John Baez is giving a series of 5 lectures at the Corfu QG School.