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:

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.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.

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

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.