Decoupling limit in AdS/CFT

Question

In the heart of the original derivation of the AdS/CFT is the decoupling limit. From the Supergravity point of view (\(g_s N \gg 1\)), say for D3 branes, this amounts to taking the near horizon limit \(r/R \ll 1 \) with \(R\) the AdS radius.  (See the well known review)

• Is there any paper that discusses different ways of taking a decoupling limit? (I am aware of the non-conformal case of other D-branes with running dilatons) ex: http://arxiv.org/pdf/hep-th/9802042.pdf
• For example, if one considers more elaborate constructions of D-brane configurations -ex compact say spherical or toroidal (Say we put a stack of N spherical D-branes together) one can in principle form a lot of dimensionless ratios that one can tune, is the obstacle finding SUGRA solutions with these geometric properties - or are in general these solutions  unstable?

Comments

This is not really an answer, but here is what Lumo said when I asked him about this questions.

Answer 1

Well, i totally agree that i did not give a very well posed problem, but I was just curious if other people had specific examples...(I have not yet tried to work out this problem myself)

Anyway, what I get from Lubos answer is that in principle one might be able to do this but of course he needs to choose a specific configuration that is stable and then discuss if this is interesting...

Personally, my original motivation (these are my thoughts so they might be totally wrong..) was that when one takes the decoupling limit the radius of AdS and of the sphere are the same and one cannot tune their ratio.... Thus we need an extra parameter to do this and I was wondering whether it can come from other brane configurations...

For me as far as I understand AdS/CFT  branes are non perturbative objects but their effects scale like $ e^{\frac{1}{g_s}} $ , so gravitationally they are not "enough" to create bh as single objects...

So we put a lot of them N and when we both take N and g to be large then they collapse and form the black hole.

Apparently, since we want a stable configuration etc we take an extremal bh and the near horizon limit is the well known $ AdS_5 \times S^5 $.

Now if one could cook up a stable configuration having an extra parameter (this is what I do not know how to do and what role do different configurations play in their near horizon geometry) I believe that in principle one can get a near horizon limit with a small sphere compared to ads.......(so this parameter to tune the relative size of the two)