One precise example of realization of the holographic principle is the CS/WZW correspondence, which relates 3d Chern-Simons theory with the 2d Wess-Zumino-Witten model. As explained for example in this article in nlab there is a relation between this correspondence and the AdS3/CFT2 correspondence (which in turn relates 3d gravity and 2d conformal field theory). In particular CS/WZW appears to be some part (nlab says "sector") of the full AdS/CFT correspondence.

How does CS/WZW appear inside AdS3/CFT2? I'm asking, if possible, for a rough explanation of how this works. It seems that this happens in other dimensions too. Is there any similar TFT/CFT correspondence that can be seen as a sector of the usual AdS/CFT setting with $\rm{AdS}_5\times S^5$ type II supergravity / 4d $\mathcal{N}=4$ super Yang-Mills?


2 Answers 2


Three Dimensional Chern-Simons Bulk with Two Dimensional Wess-Zumino-Witten Boundary is an excellent example of holography -- although it was formulated before holographic principle was recognized, but holography does not necessarily make it an AdS/CFT. The two are distinct, where AdS/CFT is a specific case of holography, which is not usually attributed to CS/WZW. There is an equivalence between the conformal blocks of two-dimensional WZW and the hilbert space of the dual three-dimensional CS theory [Witten 89]. That is not to say that CS/WZW cannot be realized as an AdS/CFT, however the CS theory would have to be CS gravity [Jackiw, et all. 82] although, I believe an arguement is made for this to be restricted to a perturbative regime [unknown].

In short, one would have to be studying the perturbative regime of bulk pure gravity with CS term (specifically the variation of the three-dimensional chern simons gravitational term) / chern simons gravity [Jackiw, et all. 82,03], and its boundary wess-zumino-witten (with appropriate gauge group) dual to classify CS/WZW as an AdS3/CFT2 correspondence!


Holographic duality doesn't necessarily require $AdS$ space. See, for example, the famous CS/Topological String duality. In particular, I don't think that it is possible to consider CS/WZW as $AdS_3/CFT_2$.

  • $\begingroup$ I'm aware that there are holographic dualities that don't involve AdS. CS/WZW is one example. However, it seems that AdS3/CFT2 has some sector that can be identified with CS/WZW (see the link in my question). Notice that I'm not saying that those two dualities are the same, but that there is some relatiom between them. $\endgroup$
    – coconut
    Commented Nov 17, 2016 at 18:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.