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I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that are easy to start off with.

But it seems to me that there are a lot of interesting topics that are very hard to find any reference on. When there is a reference, it is usually written by physicists for physicists and very hard to understand.

Some of the things that I am confused with are:

  1. Quantization of WZW models. Why the space of states should be $$\bigoplus_{\lambda\ \text{integrable}} H_\lambda \otimes H_\lambda^*$$ seems only to be explained by Gawędzki in his lectures Wess-Zumino-Witten Conformal Field Theory, in the book Constructive Quantum Field theory, edited by Velo and Wightmann. Gawędzki constructs a line bundle on the loop group $LG$ to take care of the ambiguity in the action, and then identifies $e^{-iS}$ as a section of that line bundle. He then goes on to use Feynman's path integral. However, already, this seems to involve some gaps to me.

  2. State-Field Correspondence, Covariance of primary fields in WZW models. One can consider a 'pair of pants' to give a incomplete argument, but how could one show that the correspondence is actually one-to-one? The covariance rules that the primary fields should satisfy are also just given as definitions of primary fields in most mathematical texts.

  3. What are 'unitary' representations for Lie algebras? It seems that this depends on the type of Lie algebra that is being dealt with.

  4. How to show CS/WZW correspondence. There does not seem to be a mathematical reference for this as well.

I would be very grateful if anyone could give me a list of references that could help me out on these, and maybe more.

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    $\begingroup$ This is a big subject, with a lot known, some questions still open, and many valuable perspectives. There certainly will not be a single best reference, and I would be surprised if any reference does a good job with >1 of the questions you listed. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 19, 2021 at 1:14

2 Answers 2

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A general reference for Chern-Simons theory written for mathematicians are the two papers by D. S. Freed:

  1. D. S. Freed: Classical Chern-Simons Theory, Part 1. Advances in Mathematics, 113(2):237–303, 1995. Preprint: arXiv:hep-th/9206021.
  2. D. S. Freed: Classical Chern-Simons theory, Part 2. Houston Journal of Mathematics, 28(2):293–310, 2002. see here.

The first part covers connected and simply connected gauge groups and the second part covers arbitrary compact Lie groups. They also contain a discussion of the WZW model.

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I'm more familiar with the physics oriented literature, so Fuchs' book Affine Lie Algebras and Quantum Groups - An Introduction, with Applications in Conformal Field Theory might suffer from the same problems you refer to. Perhaps worth a look, even thought it does not cover Chern-Simons theory.

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