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In the physics literature the Holographic Principle relates theories in the bulk and the theories in the asymptotic boundary. While the bulk theory is the 3D Chern-Simons theory, the corresponding boundary theory is the WZW CFT. I am not sure how complete the correspondence is, so I decide to test if the Wilson loop observables of 3D Chern-Simons (knot invariants) in the bulk can be computed in the WZW model. I am actually in doubt of this because how could you ever project a general knot to the boundary without losing information? Extra data (e.g. up/down crossing) must be "remembered" on the boundary, and I'm curious how that would be implemented on the CFT side.

One relevant work is [1], whose abstract promises such a construction in terms of gauge invariant composites of 2D WZW field. However, the work only works with $M^{3} = S^{2} \times S^{1}$. It does not pass to the asymptotic boundary, but instead it works with each piece of S^{2} instead. And, in the end, it only checks the results for trivial knots.

Questions

  1. How complete is the correspondence in this case (3DCS/2DWZW)?
  2. Where to find a detailed correspondence between the two theories; in particular, a formula for link invariants on the CFT side.

Reference

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  • $\begingroup$ I think that questions about holography are better suited for Physics.SE, so probably better to try there $\endgroup$
    – Jules Lamers
    Commented Nov 28, 2023 at 16:12
  • $\begingroup$ I seem to remember that Costello--Witten--Yamazaki look at Wilson loops in (4d?) Chern--Simons are relate it to quantum integrable vertex models, as an upgrade (with spectral parameters, i.e. affine case) of Witten's old paper about Chern--Simons and the Jones polynomial. Is that perhaps relevant or interesting for you? $\endgroup$
    – Jules Lamers
    Commented Nov 28, 2023 at 16:14
  • $\begingroup$ I'm unsure about specifics, but the picture I have in my mind is that one should draw a knot as braid diagram and then stretch them to intersect the boundary. Then to each topological line operator in the bulk one assigns a point vertrex operator in the boundary CFT. The observables then should be some conformal blocks with such insertions. $\endgroup$
    – Nikita Grygoryev
    Commented Dec 20, 2023 at 21:42
  • $\begingroup$ @NikitaGrygoryev How is the braid information implemented on the CFT side? $\endgroup$
    – Student
    Commented Jan 12 at 0:30

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