0
$\begingroup$

In the context of the AdS/CFT correspondence, the coordinate $z$ of AdS in Poincarè coordinates is often identified with an (inverse) energy scale for a CFT. I don't quite understand this identification, I tried to read different reviews to find an exact proof of this fact, with no success.

The Euclidean $AdS_{d+1}$ space in Poincarè coordinates is $\mathbb{R}^{d}\times \mathbb{R}$. If the identification is correct, It seems to imply that at each point $z$ we have a CFT at a different energy. For instance, at z=0 we have a CFT in UV and at $z=+\infty$ a CFT in IR. Is this idea correct?

Can anyone clarify these two points?

$\endgroup$
1
$\begingroup$

When formulated on Minkowski space, $ds^{2}_{CFT} = \eta_{\nu \mu} dx^\mu dx^{nu}$ , the vacuum is invariant under Poincare transformations, and by virtue of conformal invariance it is also in particular invariant under a rigid scale transformation $x^{\mu} \mapsto \alpha x^{\mu}$ which simultaneously rescales the energy $E \mapsto E/\alpha$.

Identifying inverse energy with the extra dimension (labelled $z$), the most general $5$-dimensional bulk metric consistent with these symmetries is $AdS_5$ with $$ds^2 = \frac{l^2}{z^2}\left(\eta_{\nu \mu} dx^\mu dx^{nu}+dz^2\right)$$

where the rescaled $z ∼ 1/E$ so as to express the metric in terms of one free parameter, the AdS scale $l$. A trivial change of variables, $z = l^2/r$, recasts the above metric to something akin to the metric for $AdS_5 \times S^5$.

$\endgroup$
  • $\begingroup$ This is not quite satisfactory. Why is $E = 1/z$? I want to understand how the $1/z$ is related to the RG energy scale of the CFT. And, what about the second point? Is it true that for each value of $z$ there is a CFT at the energy $1/z$? $\endgroup$ – apt45 Jul 5 '17 at 13:56
  • $\begingroup$ @apt45 Editeed my post, is this now sufficient? $\endgroup$ – Kevin Jul 5 '17 at 14:05
  • $\begingroup$ Thanks you but still it is not a proof. It would be better to show that this identification actually arises when one considers RG flows of conformal theories (at the boundary) deformed by relevant operators. I am trying to figure out this fact. Thank you anyway. So, is it true that for any $z$ we have a CFT at a given energy in $\mathbb{R}^d$? $\endgroup$ – apt45 Jul 5 '17 at 14:53
1
$\begingroup$

Since energy is propotional to $1/t$ and if we think of cft as lying on the boundary of Ads space, then the cft lives on $z = 0 $ hypersurface. Then consider a local bulk process with energy $E$ then from $d\tau = (l/z)\, dt$, we would have $$E_{CFT} = \left(\frac{l}{z}\right) E$$ since $E\sim 1/t$ and if we consider $E_{CFT}$ corresponds to the coordinate $t$ or asymptotic time. Then we can see the CFT energy is propotional to the inverse of $z$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.