I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. 


I guess in the condensed matter physics literature this is the "same" thing which is referred to when they say that one has propagating chiral bosons on the boundary of the manifold if there is a Chern-Simons theory defined in the interior("bulk") 




---


Let me quote (with some explanatory modifications) from two papers two most important aspects of the relationship that is alluded to, 


1. *"...It is well known that any Chern-Simons theory admits a boundary which carries a chiral WZW model; however these degrees of freedom are not topological (the partition function of Chern-Simons theory coupled to such boundary degrees of freedom depends on the conformal class of the metric on the boundary)..."*
2. *"...In general if the pure Chern-Simons theory (of group $G$) at level k is formulated on a Riemann furface then the number of zero-energy states equals the number of conformal blocks of the WZW model of $G$ at level $k' = k - \frac{h}{2}$ ($h=$the quadratic Casimir of G in the adjoint representation)..(and when the Riemann surface is a torus) the number of conformal blocks is equal to the number of representations of $\hat{G}$ at level $k'$.."*




---


* I would like to know of a reference(s) (hopefully pedagogical/introductory!) which explains/proves/derives the above two claims. (..I looked through various sections of the book by Toshitake Kohno on CFT which deals with similar stuff but I couldn't identity these there..may be someone could just point me to the section in that book which may be explains the above claims but may be in some different garb which I can't recognize!..)



The Chern-Simons/Wess-Zumino-Witten correspondence 
Quantum field theories are understood/formalized at various levels of detail (e.g. action functional only, space of states/partition function only, full functorial QFT, full extended QFT). Accordingly there are such different levels at which people will say "It is well-known that...".


For the general [holographic principle](http://ncatlab.org/nlab/show/holographic+principle) there are still lots of gaps, but for the special case of 3dChern-Simons-TQFT/2dWZW-CFT things are pretty well understood. 


The nLab entry 


[holographic principle -- Ordinary Chern-Simons theory / WZW-model](http://ncatlab.org/nlab/show/holographic+principle#OrdinaryCSWZWModel)


gives a list of pointers, some of which coincide with what is being said in other replies here.


First of all there is a direct relation between the action functionals: the CS action functional on a manifold with boundary is not gauge invariant. The boundary term that appears is the action functional of the WZW model (the topological term, at least, and the kinetic term with due fine-tuning).


More abstractly, the Chern-Simons action for $G$ simply connected arises by transgression of a differential universal characteristic map on higher smooth moduli stacks $\mathbf{B}G\_{conn} \to \mathbf{B}^3 U(1)\_{conn}$. The WZW action (the topolological term) similarly arises simply by the (differentially twisted) looping (as smooth $\infty$-stacks) of this map.


Then the famous original observation: geometric quantization of this action functional yields a space of states for Chern-Simons theory that may be naturally identified with the partition function of the WZW model. 


To promote this further to a relation between ful QFTs, one needs to know what the full QFT corresponding to Chern-Simons theory is. This hasn't as yet been fully established via quantization, but the expectation is that it is what the Reshetikhin-Turaev construction gives when fed the modular tensor category of loop group representations of the gauge group. Assuming this, there is a very detailed construction by Fuchs-Runkel-Schweigert and others that effectively construct the rational WZW CFT (as a full Segal-style CFT) from the TQFT.


Recently the holographic aspect of this construction has been further amplified by Kapustin-Saulina and then by Fuchs-Schweigert-Valentino.


See at the [above link](http://ncatlab.org/nlab/show/holographic+principle#OrdinaryCSWZWModel) for references to all these items.


