References for Understanding Minahan's N=4 SCFT review

Question

This is about the same paper as this thread: Some questions about chapter I.1 (by Minahan) of the "Review of AdS/CFT Integrability" but it was never answered.

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.

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user gn0m0n

Comments

@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.

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user gn0m0n

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It's also remarkable that Physics.SE keeps on getting the same question about how scalars transform under the conformal group for more than two years now.

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user Vibert

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I'd never picked up that the generators together form these kinds of matrices. I'd just gotten used to numbering them like $\sigma^i$, $i= 1, 2, 3$ or for $SU(3)$, $T^a$ with $a=1,...,8$ - or for the Lorentz generators, the $\mu , \nu$ indices are easy to understand when they are presented as $\sim$[$\gamma^\mu , \gamma^\nu$]. I don't think I've written down an $SO(N)$ basis since I was still living in 3 dimensions and talking about $SO(3)$, and more likely to talk about $J^1$ than $J^{23}$. But I see via that now how to understand these guys. thx thx

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user gn0m0n

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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

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user gn0m0n

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one year later, this question is embarrassing

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user gn0m0n

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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.

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user Luboš Motl

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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.

This post imported from StackExchange Physics at 2015-06-23 15:16 (UTC), posted by SE-user Luboš Motl