Reference request for TQFT, functoriality

Question

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.

It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;

The function $(M, \partial_{-}M, \partial_{+}M) \mapsto \tau(M): T(\partial_{-}M) \to T(\partial_{+}M)$ extends the modular functor $T$ to a non-degenerate topological quantum field theory.

The proof of the functoriality is unclear for me. I tried to look at Turaev's papers but its harder to understand. Also I don't understand the proof of computation of annomalies (Theorem 4.3 on chapter 4). The method of the proof seems to extend the method of the proof of functoriality to 4-manifolds.

Could you suggest me a textbook or paper etc that explain these theorems or similar material?

Or could you show me more detailed proof of functoriality (and the computation of anomalies) here?

Answer 1

Turaev himself has reviewed this construction of a 3-dimensional TQFT in the article Quantum 3-Manifold Invariants (2006). As "further reading" he suggests the 2001 lecture notes by Bakalov and Kirillov, which are published by the AMS but can be freely downloaded in a non-final form from here. Chapters 4 and 5 address the Turaev-Reshetikhin construction. A quick look suggests that these lecture notes are quite gentle on the reader, and may be along the lines of what you are looking for.

These lectures are devoted to the discussion of the relation between tensor categories, modular functor, and 3D topological quantum field theory. They were written as a textbook; all the results there are known. Our only contribution is putting it all together, filling the gaps, and simplifying some arguments.