It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference talking about the inverse step: how could we get a TQFT given a representation of the mapping class group of a surface S? How should I think this problem?
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1$\begingroup$ I'm not certain any representation of a surface group leads to a TQFT. There might be some restrictions, and certainly one needs a representation for each mapping class group of surfaces of all genus. $\endgroup$ – Ian Agol May 24 '12 at 23:04

8$\begingroup$ If (1) you have a collection of unitary representations of the MCG for all surfaces, and (2) these representations behave well w.r.t. gluing, and (3) the (semisimple) categories associated to stacking annuli have finitely many classes of minimal idempotents, then you have what is called a "modular functor" (or a sum of such), and standard results say that you can reconstruct the 3dimensional part of the TQFT from this. $\endgroup$ – Kevin Walker May 25 '12 at 1:30

3$\begingroup$ To add a reference to Kevin Walker's comment, modular functors and how to reconstruct a TQFT from such are discussed in Bakalov and Kirillov's book: math.sunysb.edu/~kirillov/tensor/tensor.html $\endgroup$ – Daniel Moskovich May 25 '12 at 1:36
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The following paper answers your question precisely, I think: arxiv:1408.0668
Specifically Theorem 1.3 (which is elaborated on in Section 4) describes precisely what additional data you must specify to construct a TQFT in addition to representations of the MCG of all genera. Roughly, it is the ability to "glue" together representations of different genera in a coherent manner.

$\begingroup$ Since you posted this answer, Juhász updated his paper; Theorem 1.3 in v4 is Theorem 1.10 of the current version. $\endgroup$ – Arun Debray May 16 '18 at 18:50