I've been inspired by the string-net condensation paper by Levin and Wen (see here) to learn about the Turaev-Viro model and its relationship to topological QFTs. Any references that could be helpful for accomplishing that would be very welcome. While I'm at it, pedagogical reviews/introductions to topological QFTs would be useful as well. Textbook and paper recommendations are both welcome.
I should also mention that I'm definitely not a mathematician. I'm familiar with the details of non-abelian gauge quantization, and with it the basics of representation theory. I've also made some progress learning about CFTs from the big yellow book (di Francesco et al.). But beyond that, my pure math background is a bit limited, so physically motivated arguments would be most immediately useful to me. But, if you're aware of good sources for learning about relevant math topics, that would be useful as well.