# The Chern-Simons/Wess-Zumino-Witten correspondence

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!..)
-
As far as I remember Witten's famous paper "Jones polynomial & ... " also discusses this, in the section when tries to calculate examples - at some point he needs Verlinde formula for WZ to calculate explicily something. The idea (AFAIR) is simple take locally M=R\times Sigma, main point is to choose appropriate gauge fixing, as far as I remember A_0 = 0 , then in this gauge dA_0 is Lagrange multiplier in Feynman integral which sits on the space of the flat connections... If I will remember details I will write them... –  Alexander Chervov Jul 2 '12 at 19:20

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 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

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 for references to all these items.

-
@Urs Schreiber Thanks for the details. Can you elaborate on this point you made that "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)" AFAIK because the Chern-Simons action is not gauge invariant a gauge transformation on it produces an "extra" term which looks like one of the terms of the WZW action. But I can't see how this can be interpreted to say that there is an effective WZW theory on the boundary when there is CS theory in the bulk. –  Anirbit Jul 3 '12 at 15:06
@Urs Schreiber Can you also link to the Kapustin-Saulina paper that you mentioned? (..infact the first of my italicized quotes is an adaptation from a Kapustin-Saulina paper..) –  Anirbit Jul 3 '12 at 15:08
Hi Nairbit, my comment above is really just an extended pointer to what I have written at that nLab entry, which contains the links that you are looking for and more: ncatlab.org/nlab/show/holographic+principle#OrdinaryCSWZWModel . –  Urs Schreiber Jul 6 '12 at 12:20
Sorry, my fingers introduced a key twist. I meant to type Anirbit. Sorry. (Wasn't there once the possibility to edit comments here? Where did it disappear to?) –  Urs Schreiber Jul 6 '12 at 12:21
@Urs Schreiber Thanks for the details. But don't see anywhere in your links a derivation of the fact that about the bulk partition function of Chern-Simons' theory generically having a dependence on the conformal class of the metric on the boundary (..and how specific boundary conditions for the gauge field can remove that dependence..) This to my mind is one of the most important aspects of the correspondence. Can you give some references/explanations towards that? –  Anirbit Jul 11 '12 at 15:14

The boundary of a Chern-Simons theory carries a Wess-Zumino-Witten model...

This comes from the following relation between the parameters of the two theories. Recall that a Chern-Simons theory is determined by an element $$\xi \in \hat H^4(BG,\mathbb{Z}),$$ an element in the degree four differential cohomology of the classifying space of the gauge group. Often $\xi$ can be identified with an element in ordinary cohomology, and in turn with just an integer, called level. Recall that a Wess-Zumino-Witten-model is determined by an element $$\eta \in \hat H^3(G,\mathbb{Z}).$$ Now, there is a transgression map $$t: \hat H^4(BG,\mathbb{Z}) \to \hat H^3(G,\mathbb{Z})$$ which converts a Chern-Simons theory into a WZW model.

This is discussed (using bundle gerbes) in

The states of the CS theory form the conformal blocks of the WZW model...

This is a result of Witten, a crucial ingredient for the relation between Chern-Simons theory and the Jones polynomial. You might want to start in Section 5 of

• E. Witten: Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121,351-399 (1989)

Another source with general information about the Chern-Simons states is Section 5 of

They key information is formula (5.15) in the latter paper. It expresses the partition function of the WZW model (coupled to a gauge field, and with field insertions) by scalar products of CS states. The next formula (5.16) has (5.15) reduced to the torus, relating it to representations of $G$.

-
The main derivation though of what you wrote is from the reference I posted. –  Chris Gerig Jul 2 '12 at 21:34
@Konrad Waldorf Thanks for the answer. Seems Alexander Chervov also recommended section 5 of this paper of Witten's. About Gawedski's lectures - probably its a bad question to ask! - how far back do I have to start to understand equation 5.15? (...earlier my experience has been that Gawedski's writings can be very terse!..I am just wondering whether I have to read that entire lecture to understand this point!..) –  Anirbit Jul 3 '12 at 15:18
@Anirbit: the paper of Gawedzki's is a review about certain standard aspects of conformal field theory. If you want to understand the formula, I'm afraid you will either have to read the whole paper, or to go to other sources that help you to understand at least Section 5. Gawedzki's writing might be terse, but it's also a landmark of elaborate, substantiated, and correct writing! –  Konrad Waldorf Jul 3 '12 at 23:28
@Konrad Waldorf I guess I will first look through the section 5 of Witten's paper and then will venture into Gawedski's writings. (..experience has been that Witten's writings are much more beginner friendly!..) –  Anirbit Jul 4 '12 at 20:11

The immediate paper that comes to mind is Topological Gauge Theories and Group Cohomology by Dijkgraaf and Witten, starting on pg403. The Wess-Zumino term appears because the Chern-Simons functional is not gauge invariant, and the variation of this action depends on the connection at the boundary surface. This paper references Witten's Non-abelian Bosonization in Two Dimensions in talking about the WZW model and CFT, so I think this would be useful to check out.

As for the second comment, pg411-413 brings up the conformal block stuff, but I'm not sure it explains what you want. It does have references:
1) Extended Chiral algebras and modular invariant partition functions (Karpilovsky, et.al.)
2) Spectra of WZW models with arbitrary simple groups (Felder, et.al.)
3) Taming the conformal zoo (Moore, Seiberg)
Hopefully one of those leads you to what you desire.

-

You are unlikely to find a proof of these claims, because Chern-Simons theory, as a quantum field theory in 3 dimensions, has not been precisely formulated mathematically.

You can find some partial results in the book Bakalov, Kirillov, Lectures on tensor categories and modular functor.

-
More precisely, what has not been carried out precisely is the quantization of the Chern-Simons Lagrangian to a full TQFT. On the other hand, it is expected that once done, the result is the TQFT defined by the modular tensor category given by, say, the loop group representations of the given loop group. Under this assumption, the correspondence CS-TQFT / WZW-CFT has been made precise by Fuchs-Runkel-Schweigert et al. see ncatlab.org/nlab/show/FFRS-formalism . –  Urs Schreiber Jul 3 '12 at 7:53

There is yet one more perspective on the relation between $G$-Chern-Simons theory and the WZW-model on $G$: the background B-field of the latter can be regarded as being the prequantum circle 2-bundle in codimension 2 for a "higher/extended geometric quantization" of Chern-Simons theory.

This is spelled out a bit at

In brief the story is this:

We have constructed in Cech cocycles for differential characteristic classes a refinement of the generator of $H^4(B G, \mathbb{Z})$ to a morphism of smooth moduli $\infty$-stacks $\mathbf{c}_c : \mathbf{B}G_c \to \mathbf{B}^3 U(1)_c$ from that of $G$-principal bundles with connection to that of circle 3-bundles (bundle 2-gerbes) with connection

(for $G$ a simple, simply connected Lie group).

This is such that when transgressed to the mapping $\infty$-stack from a closed compact oriented 3d manifold $\Sigma_3$ it yields the Chern-Simons action functional

$$\exp(2 \pi i \int_{\Sigma_3} [\Sigma_3, \mathbf{c}_{conn}]) : CSFields(\Sigma_3) = [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,.$$

But one can similarly transgress to mapping stacks out of a $0 \leq k \leq 3$-dimensional manifold $\Sigma_k$. For $k = 1$ with $\Sigma_1 = S^1$ one obtains a canonical circle 2-bundle (circle bundle gerbe) with connection on the smooth moduli stack of $G$-principal connections on the circle

$$\exp(2 \pi i \int_{S^1} [S^1, \mathbf{c}_{conn}]) : [\Sigma_1, \mathbf{B}G_{conn}] \to \mathbf{B}^2 U(1) \,.$$

Now since $\mathbf{B}$ is "categorical delooping" while $[S^1, -]$ is "geometric looping", the mapping stack on the left if not quite equivalent to $G$ itself, but it receives a canonical map from it

$$\bar \nabla_{can} : G \to [S^1, \mathbf{B}G_{conn}] \,.$$

In fact, the internal hom adjunct of this map is a canonical $G$-principal connection $\nabla_{can}$ on $S^1 \times G$, and this is precisely that from def. 3.3 of the article by Carey et al that Konrad mentions in his reply.

So the composite

$G \to [S^1, \mathbf{B}G_{conn}] \stackrel{transgression}{\to} \mathbf{B}^2 U(1)_{conn}$

is thw WZW circle 2-bundle on $G$, or equivalently the Chern-Simons prequantum circle 2-bundle in codimension 2.

(The math parser here gets confused when I type in the full formulas. But you can find them at the above link).

-

I learned this correspondence from Bos and Nair's paper "Coherent State Quantization of Chern-Simons Theory" and that is what I recommend. I wrote a short review of the part that you are interested in: check the second section of my paper (http://arxiv.org/abs/1311.1853) on Yang-Mills-Chern-Simons theory. Constant k comes from the wave-functional and quadratic casimir in the adjoint representation comes from the gauge invariant integral measure. Some knowledge on geometric quantization is required. You can find that in V.P.Nair's "Quantum Field Theory" book (or B.C.Hall's "Quantum theory for mathematicians" book in more detail). Also in Nair's book you can check the WZW section(17.6). It explains where quadratic casimir comes from in the "Dirac determinant in two dimensions"(17.7) calculation.

-