Question

The Ads side of the AdS/CFT correspondence is a model of quantum gravity in 5 dimensional antidesitter space. What can it say about quantum gravity in our 4-spacetime dimensions? Or is it just a toy model to better our understanding of quantum gravity as a whole?

Answer 1

The holographic priciple, as I understand it, is a general feature gravity has to obey due to its non-local behavior (as seen in the black hole entropy). These features should be there in any consistent theory of quantum gravity, and in this sense are more "general" than just string theory. It can in principle be formulated on any background, and the dual QFT has to have the asymptotic symmetries of the chosen background (asymptotically AdS has conformal symmetry).

The original Manldacena proposal is actually in 10-dimensions (critical superstring theory) where 5 of them are compactified (I think that the flux of higher form fields in the compactified dimensions play an important role, so its not enough to only keep the non-compact dimensions). Furthermore the duality can be generalized to other dimensions, 5 is not that special in this regard. There is for example a lot of work on AdS3/CFT2 holographic dualities these days.

But the crucial thing I want to say is that, the dual boundary theory is a local field theory and not a gravitational theory. Therefore the duality doesn't say too much about gravity in one lower dimension. The dual field theory "just" provides an alternative description of the (non-local) degrees of freedom in the gravitational theory.