Talk:A-D-E Classification of Conformal Field Theories
Reviewer A
I have gone over the Scholarpedia article 'A-D-E Classification of Conformal Field Theories' by Andrea Cappelli and Jean-Bernard Zuber for Scholarpedia carefully, and find it an excellent piece. The authors have done a very good and careful job. One minor point that I originally was somewhat confused about is, on further reading, explained in the clear drawings. I do not think it is necessary to change anything, since part of it is a matter of taste on how to present things. Of course, part of the presentation could have been done different, but I am happy with the way it is done now. So I recommend that the article is approved as it is.
Reviewer B:
This is an excellent review by 2 masters of the subject. The presentation is very polished. I have only a few minor suggestions/philosophic comments. I am certainly not suggesting that the authors should modify their manuscript to reflect these comments, or even that they should agree with them!
1) First, a minor point: isn't there another obvious solution to (21) than the ones tabulated in table (22)? Namely (1,q,r) for any q,r, which corresponds to graph $A_{q+r-1}$. I don't understand why this is excluded from the table. Probably related to this, I don't understand the first sentence after table (22).
2) Let's continue with this ADE theme. At the end of the paper the authors mention that ADE classifications seem to come down to 1 of 2 different basic results: either the list of symmetric matrices with largest eigenvalue <2, or the existence of triplets p,q,r of integers satisfying the inequality (21).
First of all, in my view the inequality (21) shouldn't be regarded as independent of that matrix condition, but rather as a special case of it. In particular, given any triple p,q,r obeying (21), construct the tree consisting of three strings leaving a common central vertex, of lengths p-1,q-1,r-1. There are many ways to see that such a graph (or rather its adjacency matrix) will have largest eigenvalue <2. For example, label the ith vertex from the end of the first (resp. 2nd, 3rd) string i/p (resp. i/q,i/r). Then inequality (21) is precisely the statement that $Av\le 2v$ where A is the adjacency matrix, and v is the labelling vector. Equality holds everywhere except for the central vertex. This is equivalent to the nonnegative matrix A having largest eigenvalue <2. The reverse implication, showing that any graph with largest eigenvalue <2 will necessarily be a tree corresponding to a triple obeying (21), is much less elementary. So for this reason I'd argue that (21) never really provides an independent explanation for ADE, and that the eigenvalue <2 condition (expressed either in terms of matrices or equivalently in terms of graphs) is more fundamental.
More interesting, perhaps there really is a different source for ADE classifications. Just how "different" it is, is a matter of taste. The symmetric nonneg matrices with largest eigenvalue =2 also falls into an ADE pattern (again ignoring analogues of "tadpoles"). It isn't obvious or trivial that the matrices with eigenvalue <2 should have much to do with those with eigenvalue =2 --- for example consider analogous questions for <1 and =1 or <3 and =3. (More precisely, deleting any node from a graph with eigenvalue =2 will trivially yield one with eigenvalue <2, but why should all graphs with eigenvalue <2 be obtained in this way?) But some ADE classifications are more naturally associated with <2 (for example sl(2) "nimreps") and some with =2 (for example finite subgroups of SU(2)). So for this reason, perhaps there are 2 kinds of ADE classifications.
3) It is indeed curious that the classifications for Virasoro minimal models and sl(2) chiral algebra look so alike. The explanation offered at end of paragraph containing eq.(15), is surely part of the story: the authors write something like $Vir\sim sl(2)_{p'} x sl(2)_1/sl(2)_{p'+1}$. This means the Vir minimal model reps at central charge $c=c(p'+1,p')$ are realised from affine sl(2) through the GKO coset construction. But the authors' ADE classification applies for any coprime $p,p'$, so the "unitary" case p=p'+1 is a small special case. Why should the classifications remain so similar even when (as usually would happen) |p-p'|>>1?
The ultimate explanation, I think, is still the coset realisation. When $p\ne p'+1$ the only complication is that admissible ("fractional level") reps of affine sl(2) are needed. There isn't an easy relation (that I know of) between admissible reps of level p/p' (resp. p'/p) and integrable reps of height p (resp. p'), but there is a simple relation ("Galois shuffle") between their modular matrices S,T. This relation associates each partition function for the Virasoro (p,p') minimal models to a distinct $sl(2)_p x sl(2)_{p'}$ partition function.
More generally, the same reasoning says: the modular invariant partition functions for the $W_N(p,p')$ minimal models, for any (p,p')=1, can be identified with a subset of the modular invariant partition functions for $sl(N)_p x sl(N)_{p'}$. This subset consists of all those $sl(N)_p x sl(N)_{p'}$ partition functions $N=(N_{ij;i'j'})$ satisfying some identity of the form $1=N_{J1,J'1;K1,K'1}$ for simple-currents J,J',K,K' depending on N,p,p'. The Virasoro minimal models correspond to N=2, and this extra condition is that $N_{1,p'-1;1,p'-1}=1$ for p' odd and $N_{p-1,1;p-1,1}=1$ for p' even. The details of this $W_N$ generalisation are given in Beltaos-Gannon (2010), which can be found on the arXiv.
4) The authors mention an ADE classification for N=2 superCFT minimal models. Oliver Gray's thesis, available on the arXiv, suggests it seems to me that this classification is ADE only in a very coarse grained sense, e.g. there are several N=2 superCFT minimal models, all equally realisable, and all with equal claim to the name "$E_8$".
To emphasise how well written the article is, let me end with the only typo I've found:
At end of paragraph after eq.(20), there is a "([33]." Perhaps the dangling "(" should be deleted.
and some minor notational inconsistencies:
a) 2 different abbreviations are used for trace: Tr and tr. For example, you see both used in the same paragraph on p.1.
b) $2\pi i$ vrs $2i\pi$.
c) $N_{i,j}$ with and without comma.
d) at end of paragraph containing eq.(7), $\tau\mapsto \tau+1$ etc should replace $\tau \rightarrow \tau+1$ (compare eqs(9),(10)).
