I have recently been interested in constructing topological field theories for loop quantum gravity, with a specific eye towards the topspin construction. With this in mind I was lead to the classic paper by Dijkgraaf and Witten, “Topological Gauge Theories and Group Cohomology” (1990). I am working my way through it to try and understand – which usually requires taking notes on the paper and working on things I don’t understand. I thought maybe I would post my notes here.
For an oriented, smooth 3-manifold we can specify a principal bundle with a gauge group . An action on this bundle must be constructed from some kind of invariant, to make sure it is somehow independent of the choice of connection . If is trivial, the connection is a Lie-algebra valued one-form and the action is the Chern-Simons functional
The path integral is . The integer is called the level of the theory, and must be an integer. This is because when we take to be non-abelian, under a gauge transformation the action goes to for some integer (http://arxiv.org/abs/hep-th/9902115). Thus in order for the amplitude to remain a gauge-invariant, must at least be an integer.
If is not a trivial bundle, the action is not a global gauge-invariant. However, since we know curvatures are good gauge invarients (in 4 dimensions), we can use them to define a global Chern-Simons action. So, assume we can extend the bundle over a smooth four-manifold . Then we could write
where is the usual curvature of the connection . If we have a trivial extension of the connection then the curvature is only a function of the coordinates on :
The question remains how to do this in general, since one cannot always find such an extension of the connection to the boundary (I basically chose , but we cannot guarantee that can be extended in the way that I extended ).