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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?

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@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?

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2 Answers

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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.

This post has been migrated from (A51.SE)
answered Nov 26, 2011 by Qmechanic (2,670 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.

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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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+ 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.

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answered Nov 26, 2011 by Pavel Safronov (1,115 points) [ no revision ]

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