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

Question

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?

Answer 1

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.