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I'm currently reading the paper http://arxiv.org/abs/hep-th/9405171 by Banados. I am just getting acquainted with the details of Chern-Simons theory, and I'm hoping that someone can explain/elaborate on the following statement:

In a field theory with no degrees of freedom like Chern-Simons theory, the only relevant degrees of freedom are holonomies or global charges.

I've also heard things like "Chern-Simons is topological" which sounds related. Is it?

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Is this similar to the fact that GR in 2+1 dimensions is purely topological, because vacuum curvature vanishes identically? You can have conical singularities, but not Schwarzschild-style ones. –  Ben Crowell Aug 15 '13 at 23:38
    
@BenCrowell It's certainly similar in my book, but I'm not sure to what extent these two facts are related. Chern-Simons theory involves Lie algebra valued forms on manifolds and essentially contains GR as a special case, so the statement about Chern-Simons seems a bit more general to me. –  joshphysics Aug 16 '13 at 3:27
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3 Answers 3

This is explained in Section 3 of Witten's "Quantum Field Theory and the Jones Polynomial." The idea is to locally parametrize a three-manifold by $M\times \mathbf{R}$, where $M$ is some two-dimensional manifold and $\mathbf{R}$ is the time direction that we are quantizing along. Once we do this, we can fix temporal gauge, where the time component $A_0$ of the gauge field vanishes. In this gauge, the Gauss's law constraint implies that the spatial components of the field strength vanish, which in turn says that the gauge connection is flat and the only degrees of freedom are topological.

My general feeling on Chern-Simons theory, from the limited amount that I know about it, is that most confusions that one might have are addressed in Witten's paper (unless you're interested in the relatively new field of Chern-Simons-matter.) It's a masterpiece, and also very fun to read.

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Hmm ok I'll look at Witten's paper. Thanks. –  joshphysics Mar 8 '13 at 1:40
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As far as I understand it (which is not very far but it should do), the Chern-Simons and related actions only contain what we usually call "boundary terms", so the value of the action depends on boundary conditions only and what the fields do in the bulk is irrelevant. EDIT: I had a few lectures on this this week, will post something less embarrassing after I revise properly.

Chern-Simons is topological in two senses: the first is that actions of this type only "read" topological information of the manifold on which they are integrated, as I described above. The other sense is that they are explicitly independent of any metric and volume form, so we say they are "topological" in the sense that they carry no information about the geometry of the manifold.

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Note: higher dimensional CS theories (not the 3d case) in odd dimensions have dynamical degrees of freedom, unlikely the 3d CS theory. Moreover, they are a degenerate phase-space system (in the sense of Dirac).

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