Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,852 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

References for Understanding Minahan's N=4 SCFT review

+ 1 like - 0 dislike
12 views

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
asked Apr 11, 2013 in Theoretical Physics by gn0m0n (80 points) [ no revision ]
Most voted comments show all comments
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
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
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.

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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar\varnothing$sicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...