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 , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

+ 5 like - 0 dislike
1249 views

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly.

AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ X_{-1}^2+X_0^2-X_1^2-X_2^2=L^2 $$ where $L$ is the AdS radius. Since the metric of $\mathbb{R}^{2,2}$, $$ ds^2=-dX_{-1}^2-dX_0^2+dX_1^2+dX_2^2, $$ is invariant under $SO(2,2)$ transformations and also the hyperboloid defined above is invariant we can conclude that AdS$_3$ has an $SO(2,2)$ symmetry.

One can probably show with pure group theoretical arguments that the $SO(2,2)$ symmetry is isomorphic to an $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry. I would like to know however, if one can see this symmetry more explicitly in some representation of AdS$_3$?

I suppose a starting point might be, that one can write the hyperboloid constraint equation as $$ \frac{1}{L^2}\text{det}\;\begin{pmatrix} X_{-1}-X_1 & -X_0+X_2 \\ X_0+X_2 & X_{-1}+X_1\end{pmatrix}=1 $$ i.e. there is some identification of the hyperboloid with the group manifold of $SL(2,\mathbb{R})$ itself. However, that does not tell us anything about the symmetries.

The only explanation that I have found (on page 12 of this pdf Master thesis) was that the group manifold of $SL(2,\mathbb{R})$ carries the Killing-Cartan metric $$ g=\frac{1}{2}\text{tr}\,\left(g^{-1}dg\right)^2 $$ which is invariant under the actions $$ g\rightarrow k_L\, g \qquad\text{and}\qquad g\rightarrow g\, k_R $$ with $k_L,k_R\in SL(2,\mathbb{R})$. But how does one get from the metric on $\mathbb{R}^{2,2}$ to this Killing-Cartan metric? Also, I don't find this very explicit and was wondering if there is a more direct way.

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user physicus
asked Oct 26, 2014 in Theoretical Physics by physicus (105 points) [ no revision ]
Isn't $SO(2,2)$ non-compact as well. It's algebra should be isomorphic to $so(4)$, but onlike $SO(4)$ it is non-compact, or am I wrong? Anyways, the isomorphism between $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ might be only on the level of the Lie algebras. In the literature, both are refered to being the isometry group of AdS$_3$, for example in the MAGOO AdS/CFT review page 150.

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user physicus
Related result for other signature: physics.stackexchange.com/q/28505/2451

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user Qmechanic

@physicus Do you have a good reference about the stuff you already know? Thanks!

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$\hbar$y$\varnothing$icsOverflow
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
...