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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 15th 2014

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2014
    • (edited Apr 15th 2014)

    Yeah, I saw this comment, too. Made me laugh; this is a bit like: “Is PNPP \neq NP? The nnLab 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 nnLab. 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 tmftmf-linear and be based on the GG-equivariant tmftmf-cohomology (push of BGB 3U(1)BGL 1(tmf)\mathbf{B}G \to \mathbf{B}^3 U(1) \to \mathbf{B}GL_1(tmf) ), hence GG-equivariant elliptic cohomology. That subsumes the loop group representations that André is proposing to assign to the point.