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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

4 Answers 4

up vote 23 down vote accepted

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
A belated comment, being a minor quibble on the lead-in of the above comment: Geometric quantization is pretty much what defines what one means by (non-perturbative and rigorously defined) quantization in the sense of physics/quantum field theory. It is a general mechanism that applies not just to Chern-Simons theory. On the other hand, Reshetikhin-Turaev etc. is a construction particularly designed just for Chern-Simons. If indeed RT turns out to be equivalent to GQ applied to CS, then this proves that it is indeed a good way to sum up the answer of GQ in this case. So RT better... –  Urs Schreiber Sep 3 '13 at 22:52
... So Reshetikhin-Turaev etc. better be simpler than geometric quantization, for otherwise it would be a bit besides the point. –  Urs Schreiber Sep 3 '13 at 22:53

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.

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@Jorgen May I kindly ask you to comment on the relation of this result with Drinfeld-Kohno theorem (assume genus=zero) ? KZ equation is flat connection which identifies the Hilbert spaces obtained by geometric quantization of product of orbits. (In genus zero the only parameters are "marked points z_i" -- so we need to identify Hilbert spaces obtained for different z_i). –  Alexander Chervov Feb 25 '12 at 17:41
Dear Jørgen, you say: "the equivalence of these two construction is actually known now": are you claiming that it is known for every gauge group $G$? Or are you saying that it is known for the special case $G=SU(n)$? –  André Henriques Jun 12 at 4:11
Dear Jørgen, in which paper does the skein theory model of Blanchet, Habegger, Masbaum and Vogel get identified with the modular tensor category coming from the quantum group? –  André Henriques Dec 18 at 8:26

The answer of Jørgen Ellegaard Andersen only concerns the case of when the gauge group is $SU(n)$.

I will argue that all the ingredients for the equivalence between the two approaches (namely "geometric quantization of character varieties" and "quantum groups plus skein theory") are out there, for arbitrary simply connected gauge group.

First of all, let us recall what it is that the two approaches actually construct.

$\bullet$ The first approach, developed by Axelrod-DellaPietra-Witten [1] and Hitchin [2] constructs a bundle over the moduli space of genus $g$ surfaces, along with a projectively flat connection (the so-called bundles of conformal blocks). Note that there is no way (to my knowledge) to use those bundles to construct 3-manifold invariants without, as an intermediate step, having constructed a modular tensor category.

So I'll take the point of view that "geometric quantization of character varieties" approach only constructs the bundles of conformal blocks.

$\bullet$ The second approach, by Reshetikhin and Turaev [3] takes as input the modular tensor categories coming from quantum groups (aside: the latter were only really sorted out by Sawin [4]). It produces as output a topological modular functor (which assigns vector spaces to topological surfaces with parametrized boundary, and "labels" (objects of the MTC) at each boundary components) and also a 3-dimensional TQFT (a functor from the cobordism category of surfaces and 3-dimensional cobordisms to $Vect$).

As part of the data, we also see vector spaces associated to surfaces, but this time, they are topological surfaces (equipped with a choice of Lagrangian in their first homology).

The vector spaces that appear in the second approach form part of a topological modular functor. A priori, the vector bundles that appear in the first approach are just vector bundles. But actually, they have been shown by Laszlo [5] to agree with the so-called WZW conformal blocks. The latter had in turn been shown by Tsuchiya-Ueno-Yamada [6] to satisfy factorization, i.e., to be a complex modular functor. Given the equivalence between topological modular functors and complex modular functors [7; Theorem 6.7.12], a meaningful question to ask is then:

Is the complex modular functor provided by the first approach equivalent to the complex modular functor associated to the topological modular functor provided by the second approach?

I'll take it that that's the question that the OP wanted to ask.

Andersen and Ueno [8] have showed that complex modular functors are determined by their genus zero data, and the same also holds for topological modular functors. The genus zero data is equivalent (for both complex and topological modular functors) to the data of a `weakly ribbon tensor category' [7, Theorem 5.3.8] (which will turn out to be a modular tensor category in our case of interest). So we have reduced the question to the following:

Are the modular tensor categories associated to quantum groups equivalent to the modular tensor categories coming from the WZW modular functor?

The answer is... a complicated yes: see this earlier MO question of mine.

[1] Axelrod; Della Pietra; Witten, Geometric quantization of Chern-Simons gauge theory. J. Differential Geom. 33 (1991), no. 3, 787–902.
[2] Hitchin, Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990), no. 2, 347–380.
[3] Reshetikhin; Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), no. 3, 547–597.
[4] Sawin, Quantum groups at roots of unity and modularity. J. Knot Theory Ramifications 15 (2006), no. 10, 1245–1277.
[5] Laszlo, Hitchin's and WZW connections are the same. J. Differential Geom. 49 (1998), no. 3, 547–576.
[6] Tsuchiya; Ueno; Yamada, Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure Math., 19 (1989), 459–566.
[7] Bakalov; Kirillov, Lectures on tensor categories and modular functors. University Lecture Series, 21. (All references are to the online version.)
[8] Andersen; Ueno, Modular functors are determined by their genus zero data. Quantum Topol. 3 (2012), no. 3-4, 255–291.

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Excellent. I have used your comment to boost the nLab entry a little more: ncatlab.org/nlab/show/… –  Urs Schreiber Oct 27 at 15:44

For future reference. The elegant argument sketched by Kevin Walker above involving pants decompositions and Bohr-Sommerfeld fibers was published by Andersen on the arXiv later that year, as part of the following paper:

Jørgen Ellegaard Andersen, Mapping class group invariant unitarity of the Hitchin connection over Teichmüller space, June 2012.

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