• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,720 comments
1,470 users with positive rep
818 active unimported users
More ...

  AdS/CFT in global coordinates (as opposed to Poincare coordinates)

+ 3 like - 0 dislike

When the usual derivation for AdS/CFT is given for the most famous example of Type IIB string theory on $AdS_5 \times S^5$, the AdS space is clearly seen as the near horizon geometry of a stack of D3 branes. The resulting AdS space is written in Poincare coordinates, with the Poincare horizon corresponding to the location of the D3's. For convenience, the line element of $AdS_5$ in Poincare coordinates is$ ds^2 = \frac{r^2}{L^2} dx_{\mu}dx^{\mu} + \frac{L^2}{r^2} dr^2$, and the dual field theory lives on the 4d Minkowski space.

However, AdS/CFT is quite often discussed with the AdS written in global coordinates (and sometimes in even more exotic coordinates as well). In global coordinates, the line element is $ds^2 = -\left(\frac{r^2}{L^2}+1\right) dt^2 + \left(\frac{r^2}{L^2}+1\right)^{-1} dr^2 + r^2 d\Omega_3^2$, and the dual field theory lives on $\mathbb{R}_t \times S^3$.

My question is: is there a derivation of AdS/CFT which results in global coordinates analogous to Maldacena's original derivation based on the Poincare patch?

This post imported from StackExchange Physics at 2015-01-31 12:15 (UTC), posted by SE-user Surgical Commander
asked Jan 31, 2015 in Theoretical Physics by Surgical Commander (155 points) [ no revision ]

1 Answer

+ 1 like - 0 dislike

I am not aware of a procedure to get the global AdS-metric from a decoupling limit type argument directly. You might already be aware of this, but usually the extension of the equivalence of type IIB string theory on the Poincaré patch of AdS and the super Yang-Mills theory on $\mathbb{R}^{1,3}$ to global AdS and the field theory on $\mathbb{R}\times S^3$ rests on two key observations

  1. There is a conformal embedding of $\mathbb{R}^{1,p}$ to $\mathbb{R}\times S^p$. Under the map a combination of the original generator of time translations and a generator of special conformal transformations get mapped to the the Hamiltonian (i.e. the generator of time translations along the $\mathbb{R}$ factor) in $\mathbb{R}\times S^p$. As it is a conformal mapping, the correlation functions of a CFT on $\mathbb{R}^{1,p}$ can be analytically continued to $\mathbb{R}\times S^p$.
  2. The new Hamiltonian of the boundary theory after the conformal embedding can be associated to global time translations in AdS by the matching of conformal generators on the boundary with AdS isometries in the bulk. One can see that the Killing vector field for global time translations becomes null on the Poincaré horizon and is therefore not a good time coordinate on the Poincaré patch.

The duality using global AdS is useful for several reasons, e.g. the radius of the $\mathbb{R}^3$ provides a scale against which the temperature can be measured, when we want to study the field theory at finite T. Consequently, a phase transition at finite temperature is possible (de-confinement transition) which can be associated with the transition between the AdS Schwarzschild BH and the thermal AdS geometry. In the Poincaré patch, such a phase transition cannot be seen at finite T and the field theory is always in the high temperature phase. The temperature can simply be scaled out as there is no scale to measure is against.

This post imported from StackExchange Physics at 2015-05-31 13:04 (UTC), posted by SE-user physicus
answered Feb 5, 2015 by physicus (105 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:
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