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

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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?

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André, I am not familiar with these results, but is there a Lagrangian description of these models? Or is that the question?

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@Pavel: Yes, that's exactly the question.

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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?

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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.

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answered Nov 26, 2011 by (2,860 points)
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.

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I updated the answer with a pointer to Chapter 9 in Ginsparg's lectures.

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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.

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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.

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answered Nov 26, 2011 by (1,120 points)

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