http://math.ucr.edu/home/baez/week243.html(adsbygoogle = window.adsbygoogle || []).push({});

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...in the meantime, we can think about mathematical physics. My student Derek Wise is graduating this year, and he's doing his thesis on Cartan geometry, MacDowell-Mansouri gravity and BF theory. Let me say a little about this paper of his:

11) Derek Wise, MacDowell-Mansouri gravity and Cartan geometry, available as gr-qc/0611154.

Elie Cartan is one of the most influential of 20th-century geometers. At one point he had an intense correspondence with Einstein on general relativity. His "Cartan geometry" idea is an approach to the concept of parallel transport that predates the widely used Ehresmann approach (connections on principal bundles). It simultaneously generalizes Riemannian geometry and Klein's Erlangen program (see "week213"), in which geometries are described by their symmetry groups:

Given all this, it's somewhat surprising how few physicists know about Cartan geometry!Code (Text):

EUCLIDEAN GEOMETRY -------------> KLEIN GEOMETRY

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v v

RIEMANNIAN GEOMETRY --------------> CARTAN GEOMETRY

Recognizing this, Derek explains Cartan geometry from scratch before showing how it underlies the so-called MacDowell-Mansouri approach to general relativity. This plays an important role both in supergravity and Freidel and Starodubtsev's work on quantum gravity (see "week235") - but until now, it's always seemed like a "trick".

What's the basic idea? Derek explains it all very clearly, so I'll just provide a quick sketch. Cartan describes the geometry of a lumpy bumpy space by saying what it would be like to roll a nice homogeneous "model space" on it. Homogeneous spaces are what Klein studied; now Cartan takes this idea and runs with it... or maybe we should say he rolls with it!

For example, we could study the geometry of a lumpy bumpy surface by rolling a plane on it...

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we were waiting for this TWF. Several threads here about Derek's paper and related things.

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# Baez TWF #243 is out (talks about Derek Wise Cartan paper)

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