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"Lectures on tensor categories and modular functors", Bojko Bakalov, Alexander Kirillov, 2001

Reviewed August 31, 2023

Citation: Bakalov, Bojko, and Alexander A. Kirillov. Lectures on tensor categories and modular functors. Vol. 21. American Mathematical Soc., 2001.

Web: https://www.math.stonybrook.edu/~kirillov/tensor/tensor.html

Tags: Expository, Mathematical

This is the go-to "Modular Tensor Categories" book. I don't find it to be particularly pedagogical, but it covers most of the theory and does quite a good job. Note that it "covers most of the theory" to the extent that there was a theory to cover in 2000 - there's been a lot of progress since then. Nowadays there are lots of things that are out of date about this book.

For example, they have a whole chapter on the Wess-Zumino-Witten model, which they claim is the "best known example of a modular functor". To me that sounds questionable - I only learned what the Wess-Zumino-Witten model was 5 minutes ago looking it up to write this review.

It seems like the Wess-Zumino-Witten model is the boundary theory of a Chern-Simons model. Nowadays we would just say that this is a modular functor from Chern-Simons theory and not call it Wess-Zumino-Witten. Namely, the best known example of a modular functor is now the Fibonacci theory, which is particularly important for its applications to TQC - a topic which is totally ignored in this book, because TQC was only a couple of years old.

Note that while this book certainly makes the connection between quantum groups and modular tensor categories, the exact conditions on what quantum group representation categories give modular tensor categories had yet to be worked out. This had to wait until a later work of Stephen Sawin:

> Sawin, Stephen F. "Quantum groups at roots of unity and modularity." Journal of Knot Theory and Its Ramifications 15.10 (2006): 1245-1277.