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I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D topological modular functor, where $C$ is a rigid semisimple abelian category, is 'at present unknown'. They refer to some partial results by Crane and Kohno involving Heegard splitting. They do show that a $C$-extended 2D topological modular functor gives $C$ the structure of a modular tensor category, and of course a modular tensor category gives a $C$-extended 3DTQFT, so such a construction must be possible. I was wondering whether any progress has been made in this area since the notes were written in 2000.

Many thanks in advance.

dv1
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  • I think that Andersen and Ueno claim to have an approach to something along these lines. I don't know the details well enough to know whether it is exactly what you are asking for, though. See their paper http://arxiv.org/abs/1110.5027v2 and its references, along with Andersen's MO answer: http://mathoverflow.net/questions/86792/why-hasnt-anyone-proved-that-the-two-standard-approaches-to-quantizing-chern-si/89498#89498 – Jon Paprocki Dec 09 '14 at 00:13

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