# References for Understanding Minahan's N=4 SCFT review

I have some different questions about it and I'll separate them into a couple posts if need be. I'd also be grateful if anyone can recommend other introductions or reviews for understanding N=4 generally and the Minahan review http://arxiv.org/pdf/1012.3983v2 in particular. Some of the algebra/group theory was particularly hard for me to follow (highest-weight reps, Cartan subalgebras...).

Some questions I'm particularly intrigued/troubled by are:

1. After (3.1) he says an operator $O(x)$ having dimension $\Delta$ means that when $x\rightarrow \lambda x$, then "$O(x)$ scales as $O(x) \rightarrow \lambda^{-\Delta} O(\lambda x)$." Should this be $O(x) \rightarrow \lambda^{-\Delta} O(x)$? If we say that $O(x)$ is some polynomial of degree $n$ in $x$, then after the rescaling $O(x)$ will be a polynomial of degree $n$ in $\lambda x$. So we'd have $O(x) \rightarrow O(\lambda x) \sim \lambda^n O(x)$. Then if we identify $-\Delta = n$ we have $O(x) \rightarrow O(\lambda x) \sim \lambda^{-\Delta} O(x)$. Am I missing something?

2. How does he get eq. (3.2)? It apparently follows from $D$ being the generator of scalings, by which he says he means that $O(x) \rightarrow \lambda^{-iD} O(x) \lambda^{iD}$. I'm confused by this, too, as I expect to see the generator exponentiated by $e$, not $\lambda$. I'd expect something like $e^{-i\lambda D} O(X) e^{i\lambda D}$, with $D$ as the generator and $\lambda$ as the parameter.

3. Later, in eq. (3.9), he introduces the $R_{IJ}$ as the $SO(6)$ R-symmetry generators, as well as some matrices $\sigma^{IJ}$ that he only addresses later. Here $I, J = 1...6$. I don't understand the notation. Why are there two indices on these guys? And if it's an $SO(6) \sim SU(4)$ symmetry group, then there should only be 15 generators. So are some of these $R$ and $\sigma$ redundant? Because naively it would appear that we have $6\times 6=36$ of each. I suspect that I'm missing something about how to understand these indices.

4. Kind of the same thing as 3. In (3.14) he gives some of those $\sigma^{IJ}$ and states that they are the generators in the fundamental $SU(4)$ representation. Why? Where did these come from?

I'll stop for now and post any other questions I have in another thread so as not to go overboard.

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I think these are very minor issues. 1) is about the difference between passive and active transformation. There is always a question whether you mean $\lambda$ or $1/\lambda$ and I may imagine that Joe is being sloppy here, anyway. 2) Here, $\lambda^X = \exp(X \ln\lambda)$ so if you just replace $\lambda$ by $\ln\lambda$, you relate the two expressions. The base may be $e$ but it may be anything else, too. – Luboš Motl Apr 11 '13 at 6:01
Concerning 3), generators of $SO(N)$ form an antisymmetric matrix, so the individual generators are $R_{IJ}=-R_{JI}$. Similarly, $SU(4)$ generators form a Hermitian (or anti-Hermitian) matrix. Here, you probably need a basic course on Lie groups and Lie algebras if you're asking why generators of $SO(N)$ are labeled by two indices. Similarly for 4) and $SU(N)$. Quite generally, you may be missing lots and lots of pre-requisites and you may be reading $N=4$ SCFTs too early. – Luboš Motl Apr 11 '13 at 6:04
For 3): you have surely written down a basis of $SO(N)$ (or Lorentz) generating matrices once in your life? This standard basis has naturally two indices. – Vibert Apr 11 '13 at 6:54
@lub thanks for your help. I thought that might be it for #2 but then I thought we'd be scaling by $ln\lambda$ rather than $\lambda$. Regarding #3 and #4, you may be right that I am missing pre-requisites, especially on Lie algebras as I never had a course on them - but I volunteered to talk about this topic for a sort of journal club, so, too late for now :) I'll make as much sense of it as I can. I've picked up enough group theory to understand common gauge theories and N=1 SUSY and such. But I was unfamiliar with the antisymmetric/hermitian aspect of those groups' generators. – gn0m0n Apr 11 '13 at 6:55
I see there are quite a few questions on here about SCFT and inspired by this Minahan review in particular. It looks like I am following Anirbit's footsteps so let me link to some related threads: physics.stackexchange.com/questions/4031/… , physics.stackexchange.com/questions/8634/… , physics.stackexchange.com/questions/4743/superconformal-algebra , as well as on math math.stackexchange.com/questions/23689/osp-usp-su-and-psu?rq=1 – gn0m0n Apr 12 '13 at 5:34