# Construction of 3D Topological Quantum Field Theory from a 2D Modular Functor

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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.

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