Motivation and unsolved problems of TQFT - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:14:50Z http://mathoverflow.net/feeds/question/104326 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104326/motivation-and-unsolved-problems-of-tqft Motivation and unsolved problems of TQFT Primo 2012-08-09T06:29:19Z 2012-08-09T12:26:00Z <p>I have been studying topological quantum field theory by mainly reading the Turaev's book.</p> <p>I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's hypothesis for number theory.</p> <p>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.</p> <p>Thank you.</p> http://mathoverflow.net/questions/104326/motivation-and-unsolved-problems-of-tqft/104331#104331 Answer by Daniel Moskovich for Motivation and unsolved problems of TQFT Daniel Moskovich 2012-08-09T08:11:32Z 2012-08-09T08:11:32Z <p>T. Ohtsuki's <a href="http://www.msp.warwick.ac.uk/gtm/2002/04/p024.xhtml" rel="nofollow">Problems on invariants of knots and $3$--manifolds</a> sounds to me like what you are looking for. Updates for problems in it, since it was published in 2002, are <a href="http://www.kurims.kyoto-u.ac.jp/~tomotada/solution.html" rel="nofollow">here</a>.</p> <p>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.</p> http://mathoverflow.net/questions/104326/motivation-and-unsolved-problems-of-tqft/104333#104333 Answer by André Henriques for Motivation and unsolved problems of TQFT André Henriques 2012-08-09T08:56:12Z 2012-08-09T09:07:51Z <p>The <b><i>asymptotic expansion conjecture</i></b> (AEC) states the following:</p> <p>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.</p> <p>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 <b>against</b> 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''.</p> <p><br><br><i>References:</i><br> [AH06] Andersen, Jørgen Ellegaard; Hansen, Søren Kold <i>Asymptotics of the quantum invariants for surgeries on the figure 8 knot.</i> J. Knot Theory Ramifications 15 (2006), no. 4, 479–548. </p> <p>[KSV97] Michael Karowski, Robert Schrader, and Elmar Vogt. <i>Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations.</i> Experiment. Math., 6(4):317–352, 1997.</p> http://mathoverflow.net/questions/104326/motivation-and-unsolved-problems-of-tqft/104347#104347 Answer by Kevin Walker for Motivation and unsolved problems of TQFT Kevin Walker 2012-08-09T12:09:39Z 2012-08-09T12:09:39Z <p>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.</p>