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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.)
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?
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.
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”.
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.
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,
There is also the recent book
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.
I am just repeating the full link to the reference in 7
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.
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