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:

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?

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.

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.

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