John Baez

AstroParticule et Cosmologie group, Université Paris 7

July 10, 2007

Cartan Geometry and MacDowell-Mansouri Gravity:
the Work of Derek Wise

MacDowell and Mansouri invented a clever formulation of general relativity in which the Lorentz connection and coframe field are combined into a single connection with the DeSitter group SO(4,1) or anti-DeSitter group SO(3,2) as gauge group, depending on the sign of the cosmological constant. While this formulation may seem like a 'trick', it actually has a deep geometrical meaning. This is best understood in terms of Cartan's approach to connections — an approach which was somewhat forgotten after his student Ehresmann developed the simpler approach that eventually became standard. Witten's formulation of 3d gravity as a Chern-Simons theory is also clarified using Cartan geometry. However, in 3 dimensions the relevant Cartan connection is flat and gravity is a topological field theory, while in 4 dimensions this is true only in a certain limit. In this limit, point particles and certain string-like excitations can be nicely described as topological defects. This talk is an exposition of the work of Derek Wise.

Click here to see the slides of the talk:

Much of this work is contained in Derek's thesis. For now you can read these papers of his:

Also try the transparencies of these talks by Derek:

The following papers are very relevant as well:

Text © 2007 John Baez
Images © 2007 Derek Wise