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According to this MO question – What is Chern-Simons theory expected to assign to a point? –
The nLab is somewhat vague on this subject
I wonder if this concerns Topological Quantum Field Theories from Compact Lie Groups or some other page.
Yeah, I saw this comment, too. Made me laugh; this is a bit like: “Is $P \neq NP$? The $n$Lab is a bit vague about this point.”
Anyway, giving a full description of 3d Chern-Simons as a fully extended TQFT is one of the big open questions, of which however a great deal is “in the air”. Depending on how you squint your eyes, it’s known or not. Teleman et al have been giving talks about this for a while now, one of which the above notes were taken in.
Of course eventually we should have plenty more discussion of all this on the $n$Lab. Strangely, the entries don’t write themselves.
I’ll close with just one comment: by what we have been discussing elsewhere, the quantization is naturally expected to be $tmf$-linear and be based on the $G$-equivariant $tmf$-cohomology (push of $\mathbf{B}G \to \mathbf{B}^3 U(1) \to \mathbf{B}GL_1(tmf)$), hence $G$-equivariant elliptic cohomology. That subsumes the loop group representations that André is proposing to assign to the point.
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