Question

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first published in 1991 (the geometric quantization picture here and the skein theoretic approach here), and despite a tremendous amount of development since then, it is still not known whether they are equivalent! I guess that it is reasonable to say that the problem of their equivalence has been around for 20 years.

There is (at least) one important theorem, namely the asymptotic faithfulness of the mapping class group representations produced by these two quantizations, which has proofs in both settings. The two proofs are of completely different character, and are of course logically independent, since the two representations are not known to be the same (this was proved for the quantum group skein representation by Freedman, Walker, and Wang, and for the geometric quantization representation by Andersen).

Is there a good reason why the equivalence of these two viewpoints is not yet a theorem? Is there an idea for a proof, which hasn't been completed because "it's just a long calculation" or "everyone knows it's true" or "it's nice to know, but it wouldn't actually help us prove theorems"? Or is it that it's actually a hard problem that no one knows how to approach? Is it an "important" problem whose solution would have lots of consequences and applications, or at least advance our understanding of "quantization"?

Answer 1

Good question. I'm much more familiar with the QG/skein theory approach than the geometric quantization approach, so perhaps what I write here will be biased.

I think the main reason there is not yet a proof that the two approaches are equivalent is that the geometric quantization side is difficult and unwieldy (in my biased opinion), though I'm willing to concede that it might also be beautiful and interesting. I think Jorgen Andersen has made the most progress on the GQ side, so you might want to look at his recent papers to get a feeling for what the state of the art is.

A few years ago Andersen told me the outline of an argument for proving that the two representations of the mapping class group were the same. It's a nice idea, so I'll repeat it here. I'm not sure how close Andersen and/or others are to filling in all the details.

The GQ Hilbert space for a surface $Y$ is (roughly) the space of holomorphic sections of a certain line bundle $L$ over the space of flat connections on $Y$. The holomorphic structure comes from a choice of complex structure on $Y$. Choose a pants decomposition of $Y$, and deform its complex structure by stretching transversely to the curves which define the pants decomposition. As we head toward the boundary of Teichmuller space, the holomorphic sections of $L$ will become more and more concentrated along a certain Lagrangian submanifold. In the limit, we get delta functions along this submanifold.

Recall now that instead of a complex polarization we could have chosen a real polarization; see a paper of Jeffrey and Weitsman from the early 1990's. This real polarization determines a langrangian foliation of the space of flat connections. Certain of the leaves of this foliation have trivial holonomy (of $L$); these are called the Bohr-Somerfeld orbits. Jeffrey and Weitsman showed that the number of Bohr-Somerfeld orbits matched the expected dimension of the Hilbert space.

The first punch line: The lagrangian submanifold in the Anderson picture is exactly the Bohr-Somerfeld orbits of the Jeffrey-Weitson picture. This shows that GQ gives the same answer with real or complex polarization.

The second punch line: The connected components of the Bohr-Somfeld orbits correspond to the connections where the holonomies around the pants curves take on certain discrete values in $SU(2)$. In other words, we have a finite label set (the set of allowable holonomies), and a basis of the Hilbert space for the real polarization is indexed by labelings of the pants curves by this discrete set. The skein basis is indexed by an exactly similar set of labelings of the pants curves. This gives an isomorphism between the real polarization basis and the skein basis.

I'll repeat my caveats: I'm not an expert in the above story (so I may have gotten some of the details wrong), and I think that even the experts cannot presently fill in all the details. But it seems like a nice and plausible argument to me. So far as I know it has not appeared in print, so I thought it was worth mentioning here.

Answer 2

The equivalence of these two construction is actually known now. It follows by combining the main result of Yves Laszlo: Hitchin's and WZW are the same., J. Differential Geom. 49 (1998), no. 3, 547–576 with our joint work with Kenji Ueno presented in a series of four papers:

J. E. Andersen & K. Ueno, Abelian Conformal Field theories and Determinant Bundles, International Journal of Mathematics, 18 919 - 993 (2007). J. E. Andersen & K. Ueno, Geometric Construction of Modular Functors from Conformal Field Theory, Journal of Knot theory and its Ramifications, 16 127 -- 202, (2007). J.E. Andersen & K. Ueno, Modular functors are determined by their genus zero data", arXiv: math.QA/0611087, pp. 28. To appear in Journal of Quantum Topology. J. E. Andersen \& K. Ueno, Construction of the Reshetikhin-Turaev TQFT via Conformal Field Theorem, arXiv:1110.5027

Our work with Ueno establishes the isomorphism between the Reshetikhin-Turaev TQFT for SU(n) using Quantum groups (We work with the skein theory model due to Blanchet, Habegger, Masbaum and Vogel) with the one coming from Conformal Field Theory as described in the above mentioned papers. We identify the underlying modular functors of the two theories. The natural identification of the vector space associated to a closed surface by the modular functor constructed from Conformal Field Theory with the space of covariant constant sections over Teichmüller space of the Hitchin connection, i.e. the space coming from the geometric quantization of the SU(n) charter variety is provided in the above mentioned paper with Laszlo. The composition of these isomorphisms gives the desired isomorphism.