TITLE: What is Chern-Simons theory?
QUESTION [42 upvotes]: What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics.
Chern-Simons theory is supposed to be some kind of TQFT. But what kind of TQFT exactly? When mathematicians say that it is a TQFT, does this mean that it's a certain kind of functor from a certain bordism category to a certain target category? If so, what kind of functor is it? What kind of bordism category is it? What kind of target category is it? How exactly is the functor defined?
Also, from attending talks of Michael Freeman, I know that Chern-Simons theory is supposed to describe some aspects of the fractional quantum Hall effect. How does this work? How do I take some sort of Chern-Simons computation on a 3-(or 4-?)manifold and extract from that some kind of physical prediction about some 2d electron gas? I've also heard that Witten has interpretted various knot invariants like the Jones polynomial in terms of Chern-Simons theory. So does this mean that the Jones polynomial of a knot has a physical interpretation? If so, what is it?

REPLY [2 votes]: I don't know about mathematicians but when physicist say that it is a TQFT, they just mean that the theory doesn't depend on the metric choice. You can easily see that the Chern-Simons action is metric free. Thus, it is a TQFT.
About your fractional quantum hall effect question, 
you can look at Zee's "Quantum Field Theory in a Nutshell", Part VI - Field Theory and Condensed Matter. It is a nice introduction.
About you last question, 
Witten showed that Wilson loop expectation values of Chern-Simons theory are given by link invariant polynomials. In condensed matter field theory, Wilson loop is related to conductivity. So, that is how link invariants are related to physical observables.
There are a few good books on these subjects. I recommend Altland & Simons's "Condensed matter field theory" and Fradkin's "Field Theories of Condensed Matter Physics".