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I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm looking for pedagogical reviews rather than original articles. Though these things sit at the interface of mathematics and physics I'm interested in them more as a physics student. I remember someone asking for a suggested reading list for topological QFT in mathoverflow. The suggested papers were almost uniformly rigorous mathematics written by mathematicians. I am not looking for something like that.

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5 Answers 5

The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial.

There is first of all the original article by Witten - Quantum field theory and the Jones polynomial. A related article is this one (paywall) by Elitzur, Moore, Schwimmer and Seiberg.

A very nice book is from Kauffman called Knots and Physics. Also the book by Baez and Munaiin has two introductory chapters on Chern-Simons theory and its relation to link invariants.

There are also some physical applications of Chern-Simons Theory. For instance, it appears as an effective (longe wavelength) theory of the fractional quantum Hall effect. Link invariants, such as the Jones polynomial, can be related to a generalized form of exchange statistics. See this review article: abs/0707.1889. See also this book by Lerda for more on this idea of generalized statistics.

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Thank you Olaf. I know that this stuff has very deep mathematics. However it is possible to present it without the full rigor of the professional mathematician. Anything at or below the level of Witten's papers are fine with me. In the mathematics papers that I looked into TQFT was defined using cobordism & n-category. There was nothing left to recognize it as a QFT. – Sudip Paul Oct 25 '12 at 3:14

Olaf has already given most of the references I would recommend. But in the case of Chern-Simons theories and knot theory, there are two (plus one) other very nice references. These are all written by physicist to physicists, so no modular functors, Cobordisms and so on.

1) Marcos Marino - Chern-Simons Theory and Topological Strings (arXiv:hep-th/0406005v4) Section II has a very good review of Chern-Simons theory and its relation to Knot invariants (and Rational CFT's).

2) Michio Kaku - String, Conformal Fields and M-Theory Don't be too scared by the title. Chapter 8 contain a very readable review of Chern-Simons theories and knot invariants. It introduces everything starting from a simple and intuitive starting point. For example shows how the abelian $U(1)$ Chern-Simons theory leads to the Gauss linking numbers by direct integration, and why one has to regularize with framing due to problem with self-linking. Chapter 12 is more generally about topological field theories, it discusses Cohomological Field theories, Floer theory, relations to Morse theory and so on. You might find this chapter a little more challenging than chapter 8.

3) Birmingham et al - Topological Field Theory This is a long, and a little old, review of many different topological field theories. It also contains a little bit about Chern-Simons theory but not as much as the other two above, as I remember.

I know many other good references, but they are more advanced. This is an advanced topic so most papers and books will naturally assume a certain background.

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Another interesting application is that Chern-Simons Theory in 3d is equivalent to General Relativity in 3 space-time dimensions. GR in 3 dimensions is quantisable and following a nice playground for quantum gravity. has a nice reading list about that topic at the References.

Maybe a good start is "Edward Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System Nucl. Phys. B311 (1988) 46" but if you prefer more pedagogical material i think "Bastian Wemmenhove, Quantisation of 2+1 dimensional Gravity as a Chern-Simons theory thesis (2002)" and "Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities" are very readable.

Also pedagogical to me seems "Ivancevic,Ivancevic, Undergraduate Lecture Notes in Topological Quantum Field Theory" (already linked to at your link.)

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I personally find a book by Nash Differential topology and QFT very readable.

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+1 This is a really nice book and I will also recommend this. But the book assumes the reader has a very strong background in math, much stronger than most physicist do have. So I doubt the OP will find it so useful as such. – Heidar Oct 25 '12 at 22:57
Hi Heidar, I used to be a mathematics PhD student before I transferred to physics. So I don't think that mathematical background would be an insurmountable obstacle to me. – Sudip Paul Oct 26 '12 at 7:15
@SudipPaul Oh I see, then you should definitely take a look at Nash's book. Its full of very nice algebraic/differential topology, algebraic geometry, K-theory and so on. – Heidar Oct 27 '12 at 16:41

There's a third TQFT that Witten studied in the 80s that's worth spending time with: Gromov-Witten theory, which is concerned with topological variations on the nonlinear sigma model and string theory. The starting point is Witten, Topological Sigma Models. The most recent nice exposition I know of is Hori's et al's Mirror Symmetry.

Also worth a visit is Witten, Chern-Simons Gauge Theory as a String Theory, which shows the spacetime physics of a special case of these perturturbative string theories is described by a Chern-Simons theory. Marino's book is good here, too.

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