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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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    $\begingroup$ 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
    Commented May 24, 2012 at 23:04
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    $\begingroup$ 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 (semi-simple) 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 3-dimensional part of the TQFT from this. $\endgroup$
    – Kevin Walker
    Commented May 25, 2012 at 1:30
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    $\begingroup$ 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
    Commented May 25, 2012 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.

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  • $\begingroup$ 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
    Commented May 16, 2018 at 18:50

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