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To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate on $G=SU(2)$ for the moment). Based on the perturbation analysis of the path integral "definition" of $Z_k(N)$, one predicts very precise asymptotics as $k\to\infty$, usually involving a sum/integral over the space of representations of $\pi_1(N)$ to $SU(2)$ (for instance, equation (1.32) in this paper: http://www.ams.org/mathscinet-getitem?mr=1133261)

Are any of these asymptotics rigorously proven? (the fact that the Volume Conjecture is still open leads me to think that the answer is "no")

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Note that Volume Conjecture requires a little more than a real path integral: it also requires having good complex-analytic control relating SU(2) and SL(2,C) theories. –  Theo Johnson-Freyd Aug 7 '11 at 22:49
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The most recent work that is having the most splash can be found in the preprints here: math.jussieu.fr/~marche Earlier work was done by Lisa Jeffrey and Jonathon Weitsman, Lev Rozansky, Ruth Lawrence and Don Zagier, and Jorgen Andersen. –  Charlie Frohman Aug 8 '11 at 1:50
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Don't know if this is what you're looking for, but here is a recenr preprint of Jorgen Anderson and Benjamin Himpel: front.math.ucdavis.edu/1107.1813 From the abstract: "We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups." –  Stefan Behrens Aug 8 '11 at 9:53
    
I just saw this recent paper: front.math.ucdavis.edu/1106.3066 (there's a video talk by Yamazaki at the Simons Institute). It computes the volume of punctured torus bundles in terms of the trace of an action on quantum Teichmuller space. It's not clear to me if this is related to the WRT invariants, but maybe there is a connection via work of Bonahon et al., eg: front.math.ucdavis.edu/1003.5250 Certainly from the TQFT properties of the WRT invariants, one would expect them to be related to a trace. –  Ian Agol Aug 8 '11 at 16:42
    
The papers by the groups Marche-Charles, and Jorgensen-Himpel are on the same topic. The paper by Yamazaki is on a different family of representations that come from the action on quantum Teichmuller space, which is the same set up considered by Bonahon and his collaborators, Kashaev and his collaborators and Baseilhac-Bennedetti. –  Charlie Frohman Aug 10 '11 at 16:01
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The answer is no. Rigorous results on the asymptotic expansion are known only for limited classes of 3-manifolds and/or leading terms of the expansion. See the papers cited by Charlie Frohman and Stefan Behrens in the comments to your question.

There's also a useful (though perhaps a little dated) summary of the problem by Jørgen Andersen in the 1995 Kirby problem list. (math.berkeley.edu is down at the moment, so I can't give you a verified link.) I asked Anderson recently what the state of the asymptotic expansion conjecture was, and he mentioned only the recent paper with Himpel on finite order mapping tori (see Stefan Behrens' comment).

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