# What is Chern-Simons theory?

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?

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As Ben points out below, there is already a nice nLab entry on Chern-Simons theory, answering some of my questions, though the entry is still a bit incomplete. But nevertheless, please do reply if you have any comments on things that are not covered, or not covered in detail, in the nLab entry! –  Kevin H. Lin Nov 1 '09 at 15:18

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Hmm... indeed, I should have checked nLab before posting this. Oops. –  Kevin H. Lin Oct 31 '09 at 1:18

Not sure that I am saying anything that is not in nLab, but let me try a birds eye view. A quantum theory is "mostly specified" by an action, and the CS theory in $2+1$ dimensions with group $G$ has as action the Chern-Simons functional (comes from boundary terms of characteristic classes) on the space of connections on some $3$ manifold. There is a parameter, and you get a well-defined theory whenever the parameter is a root of unity. Because this theory requires no background metric or other geometry to define, any computations in the theory should in principle be topological invariants of the manifold (life gets complicated in the details, but pretty much all you have to add is this extra biframing info to make that correct). Since particles are roughly representations, a sequence of particles interacting and moving around each other in space (a $2$-manifold here) will as a movie trace out linked loops labeled by representations in spacetime (a $3$-manifold). The expectation value of this sequence of interactions (roughly the probability of occurrence) is the value of the Jones polynomial at that value of $q$ (Those who have tried to make the various normalizations of the parameters in the physics and math literature align have gone mad: don't try it at home! It is pretty mysterious why these values combine to a polynomial). The value of the partition function for an ordinary manifold (which technically should not have physical meaning as you are supposed to divide out by it, but it is telling you something about the time-evolution operator, which is constant because it is topologically invariant).

All of that should is because the natural way to build a theory from the action is the path integral, which is a nonrigorous heuristic. The mathematical response to this is, in this case, axiomatic TQFT. Heuristic reasoning argues that the basic building blocks of the theory should have certain properties that should uniquely specify them, and then you can explicitly construct such objects from, say, quantum groups (I do NOT know how to get the quantum groups themselves from the physics) and prove they satisfy the necessary properties. From these you can compute partition functions and expectations to your hearts delight.

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@r0b0t: I think that's a bit harsh, and not tremendously polite. It's a helpful answer, and I had no trouble parsing most of it. Yes, the first sentence of the second paragraph has a rogue "should" in it, but everyone makes mistakes. –  Tom Leinster Oct 10 '11 at 21:41
@VítTuček The first sentence of the second paragraph would be fine if it had quotation marks around "should". As I understand it, the intended meaning is "All of those occurrfences of "should" in the preceding paragraph are there because ...." –  Andreas Blass Jul 23 '14 at 16:09

Some good references are the papers by Dan Freed and the book The geometry and physics of knots by Michael Atiyah. But by far the best answer to your question is in Witten's paper "Quantum field theory and the Jones polynomial", Communications in Mathematical Physics, 1989 vol. 121 (3) pp. 351-399, MR0990772.

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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".

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