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")



What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants?
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).


