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1. • CommentRowNumber1.
   • CommentAuthorBruce Bartlett
   • CommentTimeOct 13th 2012
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   Author: Bruce Bartlett
   Format: HtmlI added a query box to the <a href="http://ncatlab.org/nlab/show/holographic+principle">Holographic Principle</a> page, referring to the work of <a href="http://arxiv.org/abs/1110.5027">Andersen and Ueno</a> which I believe has now made rigorous that geometric quantization of Chern-Simons theory = quantum groups approach ala Reshetikhin-Turaev.

   I added a query box to the Holographic Principle page, referring to the work of Andersen and Ueno which I believe has now made rigorous that geometric quantization of Chern-Simons theory = quantum groups approach ala Reshetikhin-Turaev.
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 14th 2012
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   Author: Urs
   Format: MarkdownItexThanks! I have reformatted a little. (We have given up on the query boxes for questions, and in this case it is not even a question.)

   Thanks! I have reformatted a little.

   (We have given up on the query boxes for questions, and in this case it is not even a question.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 14th 2012
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   Author: Urs
   Format: MarkdownItexWait a moment. I put the reference under _[References -- RT construction and 2d CFT](http://ncatlab.org/nlab/show/holographic+principle#RTAnd2dCFT)_ -- but does the reference really derive the RT-construction from geometric quantization of the CS functional? Which theorem precisely are you thinking of?

   Wait a moment. I put the reference under References – RT construction and 2d CFT – but does the reference really derive the RT-construction from geometric quantization of the CS functional?

   Which theorem precisely are you thinking of?

4. • CommentRowNumber4.
   • CommentAuthorBruce Bartlett
   • CommentTimeOct 17th 2012
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   Author: Bruce Bartlett
   Format: HtmlI am referring to Theorem 1.1 of Andersen and Ueno's paper. How do you interpret this result? I am seeing the LHS vector spaces V_K^SU(n) as those coming from geometric quantization (sections of the line bundle over the space of fields), and the RHS vector spaces as being those coming from RT. I guess you agree with this. So I think I misinterpreted the statement, "If one accepts that the quantization of the G-Chern-Simons action functional yields the TQFT given by the Reshetikhin-Turaev construction applied to the modular tensor category of G-loop group representations, ...". In what sense is "quantization of the G-Chern-Simons action functional" murky or unknown? Do you mean that going from the Lagrangian path-integral description to the Hamiltoinan ("geometric quantization") is murky?

   I am referring to Theorem 1.1 of Andersen and Ueno's paper. How do you interpret this result? I am seeing the LHS vector spaces V_K^SU(n) as those coming from geometric quantization (sections of the line bundle over the space of fields), and the RHS vector spaces as being those coming from RT. I guess you agree with this.
   
   So I think I misinterpreted the statement, "If one accepts that the quantization of the G-Chern-Simons action functional yields the TQFT given by the Reshetikhin-Turaev construction applied to the modular tensor category of G-loop group representations, ...".
   
   In what sense is "quantization of the G-Chern-Simons action functional" murky or unknown? Do you mean that going from the Lagrangian path-integral description to the Hamiltoinan ("geometric quantization") is murky?
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2012
   • (edited Oct 18th 2012)
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   Author: Urs
   Format: MarkdownItexHi Bruce, I should say that I haven't studied the article in detail. It's a bit notation heavy and I have not had the time to sit down and plough through it. I am not doubting what you say, I am just asking so that I understand. So let me continue to ask: > I am seeing the LHS vector spaces V_K^SU(n) as those coming from geometric quantization (sections of the line bundle over the space of fields), and the RHS vector spaces as being those coming from RT. I guess you agree with this. 1. isn't it at least the other way around? The $\mathcal{V}_K^{SU(N)}$ is the modular functor described in section 4, which there is advertized as "really a generalization of the BHMV-construction of the $U_q(sl_2(\mathbb{C}))$-Reshetikhin-Turaev TQFT". 1. How do you see that the construction in section 3 is the geometric quantization of Chern-Simons theory? Just point me to the key statements. I haven't spotted them yet while browsing through this. I see geometric quantizaton mentioned in the second paragraph on p.4, where it seems to be an outlook. But that may be a wrong impression. > So I think I misinterpreted the statement, "If one accepts that the quantization of the G-Chern-Simons action functional yields the TQFT given by the Reshetikhin-Turaev construction applied to the modular tensor category of G-loop group representations, ...". > In what sense is "quantization of the G-Chern-Simons action functional" murky or unknown? What I am referring to is a perceived lack in the literature of a writeup of the following: **Task.** _For $G$ a simply connected compact simple Lie group, apply geometric quantization to the $G$-Chern-Simons action functional $\exp(i S) \colon \Omega^1(-, \mathfrak{g}) \to U(1)$ to obtain an FQFT $Bord_3 \to Vect$. Show that this functor is equivalent to the RT-construction applied to the modular tensor category $\Omega G Rep$._ Can you point me to a reference that does this completely? Maybe Andersen and Ueno do. But if so, then it's a bit hidden there. Help me extract this fully explicitly.

   Hi Bruce,

   I should say that I haven’t studied the article in detail. It’s a bit notation heavy and I have not had the time to sit down and plough through it. I am not doubting what you say, I am just asking so that I understand.

   So let me continue to ask:

   I am seeing the LHS vector spaces V_K^SU(n) as those coming from geometric quantization (sections of the line bundle over the space of fields), and the RHS vector spaces as being those coming from RT. I guess you agree with this.

   1. isn’t it at least the other way around? The $\mathcal{V}_K^{SU(N)}$ is the modular functor described in section 4, which there is advertized as “really a generalization of the BHMV-construction of the $U_q(sl_2(\mathbb{C}))$-Reshetikhin-Turaev TQFT”.

   2. How do you see that the construction in section 3 is the geometric quantization of Chern-Simons theory? Just point me to the key statements. I haven’t spotted them yet while browsing through this.

      I see geometric quantizaton mentioned in the second paragraph on p.4, where it seems to be an outlook. But that may be a wrong impression.

   So I think I misinterpreted the statement, “If one accepts that the quantization of the G-Chern-Simons action functional yields the TQFT given by the Reshetikhin-Turaev construction applied to the modular tensor category of G-loop group representations, …”.

   In what sense is “quantization of the G-Chern-Simons action functional” murky or unknown?

   What I am referring to is a perceived lack in the literature of a writeup of the following:

   Task. For $G$ a simply connected compact simple Lie group, apply geometric quantization to the $G$-Chern-Simons action functional $\exp(i S) \colon \Omega^1(-, \mathfrak{g}) \to U(1)$ to obtain an FQFT $Bord_3 \to Vect$. Show that this functor is equivalent to the RT-construction applied to the modular tensor category $\Omega G Rep$.

   Can you point me to a reference that does this completely? Maybe Andersen and Ueno do. But if so, then it’s a bit hidden there. Help me extract this fully explicitly.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeOct 17th 2012
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   Author: zskoda
   Format: MarkdownItexThe fact that Ueno is one of the authors is quite a good recommendation. He was pioneer with Tsuchiya and Kanie in very explicit description of ideas of CFT and conformal blocks for a number of special models, using explicit representation theory of affine Lie algebras and so on. See e.g. the historical reference which I listed under [[conformal blocks]], which is still a great reading for graduate students, * A. Tsuchiya, K. Ueno, Y. Yamada, _Conformal field theory on universal family of stable curves with gauge symmetries_, Adv. Studies in Pure Math. __19__, 459--566, Academic Press (1989) [MR92a:81191](http://www.ams.org/mathscinet-getitem?mr=92a:81191) There is also the recent book * Kenji Ueno, _Conformal Field Theory with Gauge Symmetry_, Fields Institute Monographs 2008 [book page](http://www.ams.org/bookstore-getitem/item=FIM-24)

   The fact that Ueno is one of the authors is quite a good recommendation. He was pioneer with Tsuchiya and Kanie in very explicit description of ideas of CFT and conformal blocks for a number of special models, using explicit representation theory of affine Lie algebras and so on. See e.g. the historical reference which I listed under conformal blocks, which is still a great reading for graduate students,

   • A. Tsuchiya, K. Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Studies in Pure Math. 19, 459–566, Academic Press (1989) MR92a:81191

   There is also the recent book

   • Kenji Ueno, Conformal Field Theory with Gauge Symmetry, Fields Institute Monographs 2008 book page
7. • CommentRowNumber7.
   • CommentAuthorBruce Bartlett
   • CommentTimeOct 17th 2012
   • PermaLink
   Author: Bruce Bartlett
   Format: Text> isn’t it at least the other way around? Yes, sorry, I meant to write it the other way round. > How do you see that the construction in section 3 is the geometric quantization of Chern-Simons theory? Fair enough, it's quite opaque in this paper. But Andersen gave a much clearer treatment in the introduction of his latest paper, http://arxiv.org/abs/1206.2785. If I read that introduction, even though I don't understand the technical details it convinces me he is really dealing with the honest-to-goodness vector space formed by geometric quantization from the Chern-Simons functional as you want. At least, he connects his work to all the previous stuff on the subject. Zoran - thanks for the references.

   > isn’t it at least the other way around?
   
   Yes, sorry, I meant to write it the other way round.
   
   > How do you see that the construction in section 3 is the geometric quantization of Chern-Simons theory?
   
   Fair enough, it's quite opaque in this paper. But Andersen gave a much clearer treatment in the introduction of his latest paper, http://arxiv.org/abs/1206.2785. If I read that introduction, even though I don't understand the technical details it convinces me he is really dealing with the honest-to-goodness vector space formed by geometric quantization from the Chern-Simons functional as you want. At least, he connects his work to all the previous stuff on the subject.
   
   Zoran - thanks for the references.
8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> much clearer treatment in the introduction of his latest paper, [http://arxiv.org/abs/1206.2785](http://arxiv.org/abs/1206.2785) Thanks, that looks good. For the moment I have just recorded this in the References-section at _[[Chern-Simons theory]]_ and _[[Reshetikhin-Turaev model]]_. Will read it in detail later when I have a second. Thanks for pointing this out.

   much clearer treatment in the introduction of his latest paper, http://arxiv.org/abs/1206.2785

   Thanks, that looks good. For the moment I have just recorded this in the References-section at Chern-Simons theory and Reshetikhin-Turaev model. Will read it in detail later when I have a second. Thanks for pointing this out.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeOct 18th 2012
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   Author: zskoda
   Format: MarkdownItexI am just repeating the full link to the reference in 7 * Jørgen Ellegaard Andersen, _A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications_, [arxiv/1206.2785](http://arxiv.org/abs/1206.2785)

   I am just repeating the full link to the reference in 7

   • Jørgen Ellegaard Andersen, A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications, arxiv/1206.2785
10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2012
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    Author: Urs
    Format: MarkdownItexAbove we discussed how the relation between the quantization of the Chern-Simons action functional to the various state sum models is strongly suggestive, but maybe not fully proven yet in the literature . Given the exchange above (#3, #4, #5 above), maybe it is worthwhile to cite the following paragraph from p. 3 of Anton Alekseev, Yves Barmaz, Pavel Mnev, _Chern-Simons Theory with Wilson Lines and Boundary in the BV-BFV Formalism_ ([arXiv:1212.6256](http://arxiv.org/abs/1212.6256)), where it says: > There is a consensus that perturbative quantization of the classical Chern-Simons theory gives the same asymptotical expansions as the combinatorial topological field theory based on quantized universal enveloping algebras at roots of unity [45], or, equivalently, on the modular category corresponding to the Wess-Zumino-Witten conformal field theory [56, 42] with the first semiclassical computations involving torsion made in [56]. However this conjecture is still open despite a number of important results in this direction, see for example[47, 3]. > One of the reasons why the conjecture is still open is that for manifolds with boundary the perturbative quantization of Chern-Simons theory has not been developed yet. On the other hand, for closed manifolds the perturbation theory involving Feynman diagrams was developed in [32, 27, 7] and in [5, 35, 13]. For the latest development see [19]. Closing this gap and developing the perturbative quantization of Chern-Simons theory for manifolds with boundary is one of the main motivations for the project started in this paper.

    Above we discussed how the relation between the quantization of the Chern-Simons action functional to the various state sum models is strongly suggestive, but maybe not fully proven yet in the literature .

    Given the exchange above (#3, #4, #5 above), maybe it is worthwhile to cite the following paragraph from p. 3 of

    Anton Alekseev, Yves Barmaz, Pavel Mnev, Chern-Simons Theory with Wilson Lines and Boundary in the BV-BFV Formalism (arXiv:1212.6256),

    where it says:

    There is a consensus that perturbative quantization of the classical Chern-Simons theory gives the same asymptotical expansions as the combinatorial topological field theory based on quantized universal enveloping algebras at roots of unity [45], or, equivalently, on the modular category corresponding to the Wess-Zumino-Witten conformal field theory [56, 42] with the first semiclassical computations involving torsion made in [56]. However this conjecture is still open despite a number of important results in this direction, see for example[47, 3].

    One of the reasons why the conjecture is still open is that for manifolds with boundary the perturbative quantization of Chern-Simons theory has not been developed yet. On the other hand, for closed manifolds the perturbation theory involving Feynman diagrams was developed in [32, 27, 7] and in [5, 35, 13]. For the latest development see [19]. Closing this gap and developing the perturbative quantization of Chern-Simons theory for manifolds with boundary is one of the main motivations for the project started in this paper.