I'm studying the holographic entanglement entropy (HEE) in this paper (Ryu-Takayanagi, 2006). In section 6.3 they compute the HEE for a segment in a 2D CFT. To do so, they obtain the corresponding geodesic in the bulk (in the Poincaré patch) and compute its length.

I understand all that process, but I'm having some trouble when they introduce the cutoff. The metric diverges when $z\to0$ so we introduce a cutoff $\epsilon>0$, I understand that. But then they say

Since $e^\rho\sim x^i/z$ near the boundary, we find $z\sim a$

Here, $\rho$ is the hiperbolic radial coordinate un the global coordinates for AdS,

$$ ds^2 = R^2(-\cosh^2\rho\ d\tau^2 + d\rho^2 + \sinh^2\rho\ d\Omega^2) $$

$x^i$ and $z$ are coordinates in the Poincaré patch,

$$ ds^2 = \frac{R^2}{z^2}(dz^2-dt^2+\sum_i(dx^i)^2) $$

And $a$ is the inverse of the UV cutoff of the CFT in the boundary, that is, the spacing between sites.

I have two problems:

1) First, I don't see why near the boundary $e^\rho\sim x^i/z$. I made up the relations between both coordinate systems and I find more complicated relations than that (even setting $z\sim0$).

2) Even assuming the previous point, I don't understand why we obtain that relation between the CFT and the $z$ cutoff.

This post imported from StackExchange Physics at 2014-09-02 07:57 (UCT), posted by SE-user David Pravos