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• CommentRowNumber1.
• CommentAuthorBruce Bartlett
• CommentTimeOct 13th 2012
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.
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 14th 2012

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.)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 14th 2012

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?

• CommentRowNumber4.
• CommentAuthorBruce Bartlett
• CommentTimeOct 17th 2012
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?
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 17th 2012
• (edited Oct 18th 2012)

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.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeOct 17th 2012

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
• CommentRowNumber7.
• CommentAuthorBruce Bartlett
• CommentTimeOct 17th 2012
> 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.
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 17th 2012

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.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeOct 18th 2012

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
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 27th 2012

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.