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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Supergroup action on $AdS_5XS^5$

+ 4 like - 0 dislike
1250 views

In the context of the AdS/CFT correspondence I was trying to understand how the symmetry group of the underlying space $AdS_5 X S^5$ comes out to be the supergroup $SU(2,2|4)$. I can see how the bosonic subrgoup $SU(2,2)XSU(4)_R$ crops up as the group of isometries of $AdS_5XS^5$, since $SU(2,2)$ is a double cover of $SO(2,4)$ and this preserves the signature (+ + - - - -)space from which $AdS_5$ arose. Moreover the group of R symmetries $SU(4)_R$ bears an astonishing resemblance to $SO(6)$ which preserves the $S^5$ metric. So far so good in bosonic land. However, what about the fermionic parts of $SU(2,2|4)$ ? How do they act on $AdS_5XS^5$ ? Something to do with the branes I guess, but I'm not sure....

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user twistor59
asked May 19, 2012 in Theoretical Physics by twistor59 (2,500 points) [ no revision ]
Dear Twistor, this is a confusing question. Fermionic generators of course don't act just geometrically on a bosonic space. At most, you could consider a superspace extension of $AdS_5\times S^5$ but superspaces are not too useful if there are too many supercharges (they have too many components). So it's a kind of misguided approach to ask about the action of the supercharges on the spacetime only; one should learn what is the action of the supergroup at the Hilbert space - the whole actual theory - and it is pretty straightforward if you define the $N=4$ gauge theory.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user Luboš Motl
Hi Lubos, the reason for the question was Witten's statement in slide number 11 in his talk in this link. He says "$AdS_5XS^5$ has symmetry group $PSU(2,2|4)$". I couldn't make sense of the geometric interpretation of the fermionic generators. Maybe he is alluding to something quantum mechanical as you say, but it wasn't clear in the slide. Also I'm not sure what the "P" means either....

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user twistor59
Dear Twistor, OK, Witten really wants to say that $PSU(2,2|4)$ is the (or "a") maximal supergroup of symmetries that a theory defined on $AdS_5\times S^5$ may have. But he surely doesn't mean that all the generators - the fermionic ones in particular - may be defined as differential operators acting on the 10 bosonic coordinates of this spacetime only.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user Luboš Motl
For the terminology of superalgebras (and the same supergroups), look at page 58 of this Kac's review (PDF): projecteuclid.org/…

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user Luboš Motl
@LubošMotl: OK thanks very much, that's clear! Post it in an answer and we can close this one off.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user twistor59
The extra $P$ in $PSU$ means that one eliminates a block-diagonal "hypercharge-like" generator from $SU$. It's similar to the embedding of $SU(3)\times SU(2)\times U(1)$ in $SU(5)$ in grand unification; in the superalgebra case, one can consistently eliminate the $U(1)$ here, at least if the number of bosonic entries and fermionic entries (dimension of the fundamental rep) are equal. And it's equal, both are 4 for $PSU(2,2|4)$.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user Luboš Motl
Ah, OK maybe P is for "projective" then.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user twistor59

1 Answer

+ 3 like - 0 dislike

Fermionic generators of course don't act just geometrically on a bosonic space; all differential operators acting on bosonic coordinates are bosonic.

At most, you could consider a superspace extension of $AdS_5 \times S^5$ but superspaces are not too useful if there are too many supercharges (they have too many components). So it's a kind of misguided approach to ask about the action of the supercharges on the spacetime only; one should learn what is the action of the supergroup at the Hilbert space – the whole actual theory – and it is pretty straightforward if you define the $N=4$ gauge theory.

Witten – when he mentions that the group acting on the AdS-space times the sphere is the supergroup – really wants to say that $PSU(2,2|4)$ is the (or "a") maximal supergroup of symmetries that a theory defined on $AdS_5\times S^5$ may have. But he surely doesn't mean that all the generators - the fermionic ones in particular - may be defined as differential operators acting on the 10 bosonic coordinates of this spacetime only.

For the terminology of superalgebras (and the same supergroups), look at page 58 of this Kac's review (PDF):

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900590

Finally, the extra $P$ in $PSU$ means that one eliminates a block-diagonal "hypercharge-like" generator from $SU$. It's similar to the embedding of $SU(3)\times SU(2)\times U(1)$ in $SU(5)$ in grand unification; in the superalgebra case, one can consistently eliminate the $U(1)$ here, at least if the number of bosonic entries and fermionic entries (dimensions of the fundamental representation) are equal. And it's equal, both are 4 for $PSU(2,2|4)$. If you check a SE question about an $SU(5)$ decomposition here

Introduction to Physical Content from Adjoint Representations

the equal dimensions allow us to set the "hypercharges" of the off-block-diagonal entries (all the fermionic generators) which were $\pm 5/6$ above to zero and eliminate the "hypercharge" $U(1)$ which was just shown to become a center (generator commuting with all others) completely.

Without the $P$ which stands for "projective", the bosonic subgroup of $SU(2,2|4)$ would really be $SU(2,2)\times SU(4)\times U(1)$ with the extra last factor that actually gets eliminated in $PSU$.

This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user Luboš Motl
answered May 19, 2012 by Luboš Motl (10,278 points) [ no revision ]

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$\hbar$ysicsOverflo$\varnothing$
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).
Please complete the anti-spam verification




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

Your rights
...