• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,721 comments
1,470 users with positive rep
818 active unimported users
More ...

  How much of the Capelli-Itzykson-Zuber ADE-classification of su(2)-conformal field theories can one see perturbatively?

+ 12 like - 0 dislike

In their celebrated work, Capelli Itzykson and Zuber established an ADE-classification of modular invariant CFTs with chiral algebra $\mathfrak{su}(2)_k$.

How much of that classification can one see using the tools of perturbative quantum field theory?

Presumably, one can't see the exceptional $E_6$, $E_7$, $E_8$ family...
what about the $A_n$ versus $D_n$ families?

This post has been migrated from (A51.SE)
asked Nov 26, 2011 in Theoretical Physics by André_1 (215 points) [ no revision ]
André, I am not familiar with these results, but is there a Lagrangian description of these models? Or is that the question?

This post has been migrated from (A51.SE)
@Pavel: Yes, that's exactly the question.

This post has been migrated from (A51.SE)
I guess you already knew things I wrote in my answer. So, can you elaborate on what exactly you mean by "the tools of perturbative QFT"? What is the theory you want to perturb around?

This post has been migrated from (A51.SE)

2 Answers

+ 4 like - 0 dislike

The following facts might be useful to OP. Firstly, the simple simply-laced $ADE$ Lie groups are in one-to-one correspondence with finite subgroups $\Gamma$ of $SU(2)$. Secondly, for $c=1$ CFTs, the $ADE$ classification is realized by orbifold models based upon modding out string propagation on a $SU(2)$ group manifold by its finite subgroups $\Gamma$, see Ref. 1 and Ref. 2, Chap.8. Thirdly, models of the $\hat{sl}(2)$ affine Kac-Moody algebra are discussed in Ref. 2, Chap.9.

Some References:

  1. P. Ginsparg, Curiosities at $c = 1$, Nucl. Phys. B 295 (1988) 153.

  2. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028.

This post has been migrated from (A51.SE)
answered Nov 26, 2011 by Qmechanic (3,120 points) [ no revision ]
Very useful indeed. I'm in the process of learning all that stuff, and references to good, comprehensive sources are always welcome. This does not, however, address the question: the theories that I mentioned above have central charge $c=3(h-2)/h$, where $h$ is the (dual) Coxeter number.

This post has been migrated from (A51.SE)
I updated the answer with a pointer to Chapter 9 in Ginsparg's lectures.

This post has been migrated from (A51.SE)
The answer here is talking about another ADE classification in CFT, which comes from the orbifolds of $SU(2)_1$ that are given by finite subgroups of $SU(2)$ (or $SO(3)$) and which are in one-to-one correspondence with ADE graphs by the McKay correspondence.

This post has been migrated from (A51.SE)
+ 4 like - 0 dislike

The $A_n$ family is obtained by considering the usual WZW action for $G=SU(2)$ and $k=n-1$. A well-known example is the free boson at $R=\sqrt{2}$, it corresponds to an $A_2$ model: its partition function is a sum of characters for spin-0 and spin-1/2 representations.

The $D_{2\rho+2}$ and $D_{2\rho+1}$ families correspond to twisting the above theories by the non-trivial outer automorphism of $\hat{A}_1$, it turns out you can do that only for even levels.

Besides Ginsparg's lectures, you can take a look at chapter 17 in Di Francesco et al. book.

This post has been migrated from (A51.SE)
answered Nov 26, 2011 by Pavel Safronov (1,120 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights