I have been studying topological quantum field theory by mainly reading the Turaev's book.

I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's hypothesis for number theory.

I also would like to know if there is a paper or book that list big or small unsolved problems of TQFT. If not, could you suggest some problems here? I have been learning TQFT but I don't know what to do by myself as a graduate student.

Thank you.

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    $\begingroup$ Ask your advisor? $\endgroup$ – HJRW Aug 9 '12 at 7:45
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    $\begingroup$ Also, this question should be community wiki. $\endgroup$ – HJRW Aug 9 '12 at 9:23

T. Ohtsuki's Problems on invariants of knots and $3$--manifolds sounds to me like what you are looking for. Updates for problems in it, since it was published in 2002, are here.

In my opinion, the biggest open problem is to relate TQFT invariants to the rest of $3$-manifold topology, one aspect of which is the Volume Conjecture.

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    $\begingroup$ +1 for mentioning the Volume Conjecture $\endgroup$ – Simon Willerton Aug 9 '12 at 8:25

The asymptotic expansion conjecture (AEC) states the following:

Let $M$ be a 3-manifold. Putting $r := k+h^{\vee}$ with $h^{\vee}$ the dual Coxeter number of the Lie algebra of $G$, the AEC states that the asymptotic expansion of the 3-manfold invariant $Z_{k}^{G}(M)$ for large $r$ would be of the form $$\sum_{j=0}^{n}e^{2\pi i\,r\,q_{j}}r^{d_{j}}b_{j} (1+\sum_{l=1}^{\infty}a_{j}^{l}r^{-l}),$$ where $d_{j}\in\Bbb{Q}$, $b_{j},a_{j}^{l}\in\Bbb{C}$, and $q_{j}\in\Bbb{R}/\Bbb{Z}$. Moreover, the set $\{q_{0}=0, q_{1},\dots,q_{n}\}$ should consist of the values of the Chern-Simons functional.

See the paper [AH06] for a survey of known results on the AEC (ok, it's 6 years old...). Note that, according to [AH06], the paper [KSV97] suggests numerical evidence against the conjectures for the 3-manifold $S^3 (4_1 ( −n/1))$, $n = 7, 16, 22$ (a manifold obtained by doing a particular Dehn surgery on a figure-eight knot), "demonstrating a contribution from a non-Chern–Simons-value phase of order $−2$ in the level".

[AH06] Andersen, Jørgen Ellegaard; Hansen, Søren Kold Asymptotics of the quantum invariants for surgeries on the figure 8 knot. J. Knot Theory Ramifications 15 (2006), no. 4, 479–548.

[KSV97] Michael Karowski, Robert Schrader, and Elmar Vogt. Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations. Experiment. Math., 6(4):317–352, 1997.

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    $\begingroup$ I think that maybe you meant to refer to (Subsection 1.2 of) arxiv.org/abs/1206.2552? That one line is from that paper anyway. Here we also include a more up-to-date short survey and mention some more recent developments on the problem. $\endgroup$ – Søren Fuglede Jørgensen Oct 25 '12 at 12:20

There are various open classification problems: classify modular tensor categories (the input for Reshetikhin-Turaev type theories), classify semisimple pivotal 2-categories (the input for Turaev-Viro type theories). There is a vague conjecture, popular among physicists, that all examples of modular tensor categories are obtainable in some way from the standard $Rep(U_q(\mathfrak g))$ examples.

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    $\begingroup$ That conjecture is made into a mathematical conjecture in the paper "The Witt group of non-degenerate braided fusion categories" by Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik. The conjecture claims that the above mentioned Witt group is generated by the modular categories coming from $U_q\mathfrak g$. $\endgroup$ – André Henriques Aug 9 '12 at 12:14
  • $\begingroup$ Thanks André. I had not realized/noticed that the DMNO work was related to things like coset constructions. $\endgroup$ – Kevin Walker Aug 9 '12 at 14:25
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    $\begingroup$ Note that DMNO only discuss the possibility that the unitary part of the Witt group, rather than the whole Witt group, might be generated by the quantum group categories. $\endgroup$ – cdouglas Aug 16 '12 at 5:57

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