Is there a reversed GKPW for AdS/CFT?

[Demystifier]

In Ads/CFT, the famous GKPW equation gives a recipe how to calculate correlators in the boundary theory by using the bulk theory. But is there a reverse? What if I want to calculate the correlators in the bulk theory by using the boundary theory?

 

[fzero]

You use the same formula. However, unlike in a QFT generating functional, where we can describe general correlation functions of off-shell objects, here we can only calculate on-shell bulk amplitudes. On the boundary, we have correlators of off-shell operators, but the sources for the bulk fields are the on-shell boundary values of the fields. Explicitly, we can say something like for an $n$-point function in the bulk

$$ \int d\mathbf{x}'_1 dr_1 \cdots d\mathbf{x}_n dr_n A_{1\cdots n}(\mathbf{x}'_1,r_1;\ldots \mathbf{x}'_n,r_n) K(\mathbf{x}'_1,r_1; \mathbf{x}_1) \cdots K(\mathbf{x}'_n,r_n; \mathbf{x}_n) =\langle \mathcal{O}_1(\mathbf{x}_1) \cdots \mathcal{O}_n(\mathbf{x}_n)\rangle,$$

where $\mathbf{x}_I$ are the boundary coordinates, $r_I$ is the radial variable and $K(\mathbf{x}'_I,r_I; \mathbf{x}_I)$ are the boundary to bulk propagators. It is not expected that one can manipulate these expressions to obtain the bare $A_{1\cdots n}$, but the amplitudes for all combinations of physical states are in principle determined from the boundary correlators.

 

[atyy]

Here are some papers about getting bulk observables from the boundary.
http://arxiv.org/abs/1102.2910
http://arxiv.org/abs/1201.3664

 

[Demystifier]

Fzero, thank you for your illuminating answer, in view of which I can reduce my question to the following one:

$K(\mathbf{x}'_I,r_I; \mathbf{x}_I)$ are the boundary to bulk propagators.

How do I compute them? I mean, can I compute them by using only the boundary theory?

 

[fzero]

How do I compute them? I mean, can I compute them by using only the boundary theory?

As usual the propagator is obtained as the Green function for the appropriate equation of motion for the field in AdS. For example, with the upper-half space metric, the bulk-boundary function for AdS$_{d+1}$ for a scalar field of mass $m$ is

$$ K_\Delta(\mathbf{x}',z;\mathbf{x}) = \frac{z^\Delta}{(z^2+|\mathbf{x}'-\mathbf{x}|^2)^\Delta},$$

where

$$ \Delta = \frac{1}{2} ( d + \sqrt{d^2+4m^2 })$$

is also the dimension of the dual operator in the boundary CFT.

 

[Demystifier]

As usual the propagator is obtained as the Green function for the appropriate equation of motion for the field in AdS.

By the field, you mean the free field, right? But what about the interacting fields in AdS?

EDIT: This problem seems to be solved perturbatively in the first paper linked by atty.