The two standard approaches to the quantization of [Chern-Simons theory](http://www.ams.org/mathscinet-getitem?mr=990772) 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](http://www.ams.org/mathscinet-getitem?mr=1100212) and the skein theoretic approach [here](http://www.ams.org/mathscinet-getitem?mr=1091619)), 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](http://www.ams.org/mathscinet-getitem?mr=1943758), and for the geometric quantization representation by [Andersen](http://www.ams.org/mathscinet-getitem?mr=2195137)).


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"?



Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The equivalence of these two construction is actually known now. It follows by combining the main result of: 



> 
> Yves Laszlo, *Hitchin's and WZW connections are the same*., J. Differential Geom. **49** (1998), no. 3, 547–576, doi:[10.4310/jdg/1214461110](http://doi.org/10.4310/jdg/1214461110)
> 
> 
> 


with my joint work with Kenji Ueno presented in a series of four papers:



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> J.E. Andersen & K. Ueno, *Abelian Conformal Field theories and Determinant Bundles*, International Journal of Mathematics, **18** (2007) 919–993 doi:[10.1142/S0129167X07004369](https://doi.org/10.1142/S0129167X07004369), arXiv:[math/0304135](https://arxiv.org/abs/math/0304135).
> 
> 
> J.E. Andersen & K. Ueno, *Geometric Construction of Modular
>  Functors from Conformal Field Theory*, Journal of Knot theory and its
>  Ramifications, **16** (2007) 127–202 doi:[10.1142/S0218216507005233](https://doi.org/10.1142/S0218216507005233), arXiv:[math/0306235](https://arxiv.org/abs/math/0306235).
> 
> 
> J.E. Andersen & K. Ueno, *Modular functors are determined by
>  their genus zero data*, Quantum Topol. **3** (2012), 255–291, doi:[10.4171/QT/29](https://doi.org/10.4171/QT/29), arXiv:[math.QA/0611087](https://arxiv.org/abs/math/0611087).
> 
> 
> J.E. Andersen & K. Ueno, *Construction of the Reshetikhin-Turaev TQFT via
>  Conformal Field Theory*, Invent. Math. **201** (2015) 519–559, doi:[10.1007/s00222-014-0555-7](https://doi.org/10.1007/s00222-014-0555-7), arXiv:[1110.5027](https://arxiv.org/abs/1110.5027)
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> 
> 


The 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 $SU(n)$ character variety is provided in the above mentioned paper with Laszlo. The composition of these isomorphisms gives the desired isomorphism.


