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:

- Cartan geometry and MacDowell-Mansouri gravity: the work of Derek Wise, in PDF.

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

- Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.
- John Baez, Alissa Crans and Derek Wise, Exotic statistics for strings in BF theory.

Also try the transparencies of these talks by Derek:

- Derek Wise, Spacetime geometry and Cartan connections, 23rd Pacific Coast Gravity Meeting, Caltech, March 16, 2007.
- Derek Wise, Exotic statistics and particle types in 3d and 4d BF theory, Perimeter Institute, July 13, 2006.

- Laurent Freidel and Artem Starodubtsev, Quantum gravity in terms of topological observables.
- Laurent Freidel, Jerzy Kowalski-Glikman and Artem Starodubtsev, Particles as Wilson lines of the gravitational field.

Text © 2007 John Baez

Images © 2007 Derek Wise

baez@math.removethis.ucr.andthis.edu