# CFT energy scale in AdS/CFT correspondence

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?

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 $AdS_5 \times S^5$.
• 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 $1/z$? – apt45 Jul 5 '17 at 13:56
• 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 $\mathbb{R}^d$? – apt45 Jul 5 '17 at 14:53
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 $$z$$.