# Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

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"?

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About representation of mapping class group - my feelings - that in genus zero (with "marked points") - it is Kohno-Drinfeld theorem - which says that monodromy representation of Knizhnik-Zamolodchikov equation is given by quantum R-matrix of the corresponding quantum group. I am not sure that higher genus is known, but actually I would say it would not be surprising... I am not big expert but I think that there no problems of seeing equivalences on the physical level of rigour... There are not only these two approaches - people worked on CS quantization by standard QFT - gauge fixing... –  Alexander Chervov Jan 27 '12 at 7:52
One more comment. There is "elder brother" for CS - quantization of Teichmuller space. The relation is like: su(2)(compact) VS. sl(2,R) -non-compact (=> more complicated is Teichmuller). More formally in CS you quantize symplectic manifold Moduli($\pi_1->SU(2)$) in Teichmuller you quantize Moduli($\pi_1->PSL(2,R)$). As far as I understand last decade the progress in quantum Teichmuller was quite big - see paper by Vladimir Fock and coauthors in arXiv. As far as I understand they can prove that all quantizations coincide for Teichmuller... It is related to "Liouville QFT". –  Alexander Chervov Jan 27 '12 at 8:16
Anderson has claimed this equivalence in talks, and used it to prove that the colored Jones polynomials distinguish knots from the unknot. But I don't think this work has appeared, I think he has made progress with collaborators though. –  Ian Agol Jan 27 '12 at 18:55
@Agol you think equivalence is not just Kohno-Drinfeld ? Why ? –  Alexander Chervov Jan 27 '12 at 19:24
@Alexander, is it known that the KZ monodromy is equivalent to the monodromy of the Hitchin connection in geometric quantization? Actually, according to an answer to a previous question of mine (mathoverflow.net/questions/73729), it's nontrivial even to relate the quantum hilbert space for the punctured sphere and the vector space on which the KZ equations act (I'd be happy to be wrong about this, though). –  John Pardon Jan 27 '12 at 20:22

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.

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Remark. Number of holomorphic sections $H^0(L^k)$ is given by the famous Verlinde formula. As Kevin mentioned this is Hilbert space of the theory in holomorphic polarization. Level "k" in Verlinde formula corresponds to $k$-th power of basic line bundle –  Alexander Chervov Jan 27 '12 at 10:05
@Kevin. Interesting ideas ! What means "stretching transversely" in the sentence about the deformation of the complex structure ? I do not quite understand what deformation you mean and why it approaches boundary... –  Alexander Chervov Jan 27 '12 at 10:08
"The lagrangian submanifold in the Anderson picture is exactly the Bohr-Somerfeld orbits of the Jeffrey-Weitson picture " ... 1) may be you need also to take classical limit k->inf ? 2) Do you mean that independently of the way to approach Teich. boundary any holomorphic section will become a linear combination of "delta"-functions concentrated along BS-tori ? Or it is somehow dependent ? Or the choice of pants selects unique way to approach boundary ? –  Alexander Chervov Jan 27 '12 at 10:16
@Alexander Chervov: In terms of the corresponding hyperbolic metric on $Y$, I mean that distances transverse to the pants curves become large and distances tangent to the pants curves become small. In the limit the curves are pinched to points and the pairs of pants get the complete hyperbolic metric of a 3-punctured sphere. I think this is a standard construction, but I'm probably using non-standard language to describe it. The level $k$ stays fixed. –  Kevin Walker Jan 27 '12 at 14:10
@Kevin thank you ! I never thought about pants, is there some reference with pictures for "easy reading" ? Hmmm if you do not need to take classical limit, but you see localization of functions ... that something new for me... If it is true there should be other cases and toy models for this phenomena... Do you know ones ? –  Alexander Chervov Jan 27 '12 at 19:29