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Citation: Balsam, Benjamin. "Turaev-Viro invariants as an extended TQFT III." arXiv preprint arXiv:1012.0560 (2010).
This is the paper which proves that the Turaev-Viro construction applied to a spherical fusion category C will be the same as the Reshetikhin-Turaev construction applies to its Drinfeld center.
This is done by constructing a nice explicit canonical natural transformation between the two functors, and proving that everything works. This is no simple task. In fact, the proof of this theorem is complicated enough that it is the culmination of a sequence of three papers, the first two of which are
> Kirillov Jr, Alexander, and Benjamin Balsam. "Turaev-Viro invariants as an extended TQFT." arXiv preprint arXiv:1004.1533 (2010).and
> Balsam, Benjamin. "Turaev-Viro invariants as an extended TQFT II." arXiv preprint arXiv:1010.1222 (2010).