It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary is given on this nLab page:


<http://ncatlab.org/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence>


This is often claimed (e.g. in that nLab page) to be an instance of the holographic principle. To me, this carries the implication that the two theories are strictly equivalent---that is, any calculation of a physical quantity in one theory could be carried out just as well in the other theory. This aspect of the holographic principle is frequently emphasized in general accounts of the holographic principle.


However, I do not see that this is delivered by the precise nature of the CS/WZW correspondence. In particular, the states of the bulk CS theory correspond only to the conformal blocks of the WZW theory, which are essentially the space of solutions to the Ward equations. While this is interesting, it is not as strong as one might expect: in particular, I don't see that it gives a way to translate some calculation in the CQFT---for example, the numerical value of a correlator for a given conformal surface with labelled marked points---into a corresponding calculation in the TQFT.


My question is the following: **If the CS/WZW correspondence is a holographic duality, why does it not seem possible to replicate every calculation from one theory in the other?**


Of course, a good answer to this question might deny the premise, or show some way in which one can indeed replicate calculations from one theory in the other theory.



Interpreting the CS/WZW correspondence
Actually I think the idea of the holographic principle is that, as in a holograph, all the information in the 'bulk' is already present at the 'boundary'. So, it claims that any calculation involving bulk observables can be expressed in terms of boundary observables. It may not claim the reverse, though that could often be taken for granted!


In discussions with Urs Schreiber and Jamie Vicary we seem to have settled on the following formulation. A modular tensor category gives rise to a once extended 3d TQFT and can also be reconstructed from this once extended 3d TQFT. By work of [Fuchs, Runkel and Schweigert](http://ncatlab.org/nlab/show/FRS-theorem+on+rational+2d+CFT), a modular tensor category *equipped with an equivalence to a category of representations of a vertex operator algebra* and *equipped with a symmetric Frobenius object* gives rise to a rational CFT.


The italicized phrases would then be ways that the rational CFT has more information than the once extended 3d TQFT. The first one gives the 'local information' needed to construct a CFT *locally* starting from the extended TQFT. The second one gives the 'global information' or 'sewing information' needed to finish the job. 


Apparently there may be one, many or no rational CFTs corresponding to a given once extended 3d TQFT.


