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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:


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.

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I am asking me the a similar question, but I am in a hurry at the moment. The point is CS can be constructed only out of the modular tensor category, while WZW also needs the VOA. –  Marcel Bischoff Mar 15 at 11:28
Indeed, that is a closely-related fact that also implies that the CQFT is richer than the TQFT. –  Jamie Vicary Mar 15 at 11:42

3 Answers 3

up vote 11 down vote accepted

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, 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.

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In AdS/CFT, at least in the best-understood variants, my understanding is that the theories on the bulk and boundary are exactly dual, and the word `duality' is often used in this context. So in this case, at least, it would be expected that each theory can be interpreted in the other. So are you suggesting that the CS/WZW correspondence is of a lesser nature, and not a duality in the same sense? –  Jamie Vicary Mar 16 at 23:13
I don't understand the AdS/CFT stuff very well, so for all I know it works the same as CS/WZW, without the precise theorems. Note you only detected the need for a bit of extra data in the CS/WZW correspondence by having a theorem relating a large class of 3d topological field theories to a large class of 2d conformal field theories. I've never heard of theorems or even precise claims like this in the higher-dimensional cases, so it could be that people are 'smuggling in' extra data on one side of the 'duality' to make the correspondence work, without knowing or caring very much. –  John Baez Mar 17 at 9:30
By the way, I dislike the trend in theoretical physics toward using 'duality' to mean a wide variety of isomorphisms, correspondences, and other relationships. To me a duality happens when we have two categories containing a dualizing object or, as a special case, a closed category containing a dualizing object - or various categorified versions of this theme. The dualities in linear and projective geometry are the prototypes. –  John Baez Mar 17 at 9:36
It's confusing that the situation in classical field theory is the opposite: a classical WZW model is a bundle gerbe with connection over the group $G$, while a classical Chern-Simons theory is a multiplicative bundle gerbe with connection over $G$, which is additional structure. In particular, there can be different multiplicative structures on the same gerbe, i.e. different TFTs corresponding to one CFT (for that, $G$ has to be non-simple or so). –  Konrad Waldorf Mar 17 at 10:06

One of the issues here is that if we define CS by its physics definition (action, fields) and similarly for WZW then it does seem like there is an equivalence. For example, on page 30 of


we apparently see how to go from a state in CS theory to a correlator in WZW. From a mathematical point of view, this uses extra information about the states in CS theory (i.e. they are not just abstract vectors in a vector space, they are functionals on flat connections). In other words, we need information about CS theory which isn't contained merely in its associated bordism representation (= mathematical notion of TQFT) $Z : Bord \rightarrow Vect$.

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The CS/WZW correspondence is of a slightly different nature than the usual AdS/CFT correspondences. First, calling the first one CS/WZW correspondence is misleading, because the "CFT"-dual is not really the WZW model, but as you pointed out, only its "chiral" part, i.e. its chiral conformal blocks. Both are instances of holography, but the real interest of the usual AdS/CFT correspondences is that the theory dual to the CFT contains gravity. Equivalently, the CFT is an honest CFT including a stress-energy tensor (which sources the graviton in the bulk), unlike the CS/WZW correspondence.

There is one setting in which they can be compared instructively. This is the case when the gauge group of the Chern-Simons theory is $SL(2,R) \times SL(2,R)$. In this case, modulo subtleties, the Chern-Simons theory is equivalent to 3d general relativity (see for instance this paper by Witten). It turns out that if you want to obtain an asymptotically AdS spacetime, you have to impose stronger boundary conditions that you would normally impose on a Chern-Simons theory. This is explained for instance clearly in Section 4 of this paper. The effects of these boundary conditions on the asymptotic symmetry algebra is to reduce the $sl(2,R) \oplus sl(2,R)$ Kac-Moody symmetry to the Virasoro symmetry (some kind of Drinfeld-Sokolov reduction).

The CFT dual to pure 3d gravity is not known (see for instance this paper and follow ups), but there have been recent conjectures (and extensive checks of these) about the AdS duals of rather simple and exactly solvable CFTs, including minimal models.

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The word "CS/WZW correspondence" seems quite appropriate given that locally the field/correlator correspondence is that of holography and globally the non-chiral WZW part is precisely encoded by the CS-theory, as for instance manifestly so in the FRS theorem (ncatlab.org/nlab/show/FRS-theorem+on+rational+2d+CFT). (Of course with two-sided boundary.) If anything deserves the word "correspondence", then this should be an example. The word "duality" on the other hand, this seems to better be avoided, both for CS/WZW as well as, I would think, for genuine AdS/CFT. –  Urs Schreiber Mar 26 at 17:07
The relation between 3d CS and 3d gravity is at least subtle. Notice that Witten essentially retracted his 1988 claim to have quantized 3d gravity in 2007 in "Three-Dimensional Gravity Revisited" (arxiv.org/abs/0706.3359). ncatlab.org/nlab/show/… –  Urs Schreiber Mar 26 at 17:09
I agree with your point about the FRS theorem. But we're now drifting even further from the genuine AdS/CFT story. Concerning the use of the word "duality", it is used in physics when two theories with different formulations turn out to be identical (in the physical sense: there is a bijection of observables, etc...). I feel it is appropriate for genuine AdS/CFT. Concerning your second comment, the discussion about the change in boundary conditions required to obtain AdS in 3d gravity requires only the classical theory, which is still subtle, but understood. –  Samuel Monnier Mar 26 at 17:41
Actually, you should maybe post your comment about the FRS theorem as an answer, it looks like it fully addresses the concerns of the OP about the CS/WZW correspondence. –  Samuel Monnier Mar 26 at 17:47

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