# Chern-Simons degrees of freedom

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?

• 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. – user4552 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
• @joshphysics I guess it is somehow clear to you that they are relevant since they are gauge independent degrees of freedom. This is related to the fact that CS theory has no local degrees of freedom (i.e, no waves,...) as can be shown by a, e.g., canonical analysis. My puzzlement (and I guess yours) is why they should be the ONLY ones, i.e., the proof of completeness. I'm unaware of a place where this is shown and the author certainly doesn't. – ungerade May 23 '19 at 16:44

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.