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While editing the tag wiki for , I initially wrote something on the lines of:

The AdS/CFT correspondence is a special case of the holographic principle. It states that a gravitating theory in Anti-de-Sitter (AdS) space is exactly equivalent to the gauge theory/Conformal Field Theory (CFT) on its boundary.

But then wondered if it would be more accurate to write:

The AdS/CFT correspondence is a special case of the holographic principle. It states that a quantum gravitating theory in Anti-de-Sitter (AdS) space is exactly equivalent to the gauge theory/Conformal Field Theory (CFT) on its boundary.

Supergravity, for instance, doesn't have a CFT dual, so which one is right? Does the theory in the AdS space have to be a quantum theory? Are there any known examples to the contrary? Can, say, general relativity on AdS be equivalent to a CFT on its boundary?

This is kind of the converse question to Which CFTs have AdS/CFT duals?

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    $\begingroup$ Every credible example of AdS/CFT that I can think of, ultimately identifies a quantum field theory (the CFT) with a string theory (or M theory) in an AdS space (ie AdS_k x M, where M is some manifold). If people say that the duality relates a gauge theory to quantum supergravity, classical supergravity, classical gravity... they are just talking about different limits of the string theory, which is the true ultimate object on the other side of the duality. $\endgroup$ – Mitchell Porter Jul 28 '13 at 1:45
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    $\begingroup$ There are various generalizations of AdS/CFT like "Kerr/CFT" or a "hydrodynamic" version which don't have a clear string-theory parent, but you would expect that in their final form, these dualities also would have quantum gravity and thus(?) string theory on the gravity side. $\endgroup$ – Mitchell Porter Jul 28 '13 at 1:46
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    $\begingroup$ The attitude I would adopt, is that every such correspondence has a quantum/quantum exact correspondence as its ultimate foundation. But I can't prove it; I'm just assuming that the situation in AdS/CFT is reproduced in these other areas, when we finally know all the facts... $\endgroup$ – Mitchell Porter Jul 28 '13 at 2:53
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    $\begingroup$ The "fluid/gravity correspondence" (which is the hydrodynamic one) arxiv.org/abs/1107.5780 is a classical/classical correspondence. The Kerr/CFT correspondence arxiv.org/abs/1203.3561 I think just relates some macroscopic properties of the black hole (like charge, mass, angular momentum) to basic parameters of the CFT (e.g. "central charge"). This is ongoing research so we don't have the full context for these relationships yet. $\endgroup$ – Mitchell Porter Jul 28 '13 at 2:56
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    $\begingroup$ @Dimension10: The correct question would be : Does holography works for classical gravity ?. The fact that the entropy of a black hole is proportionnal to its surface suggests that yes. But it does not mean that there is a quantum field theory (defined on the horizon) dual to a non-quantum gravity theory. If you take a limit like (weak energy, non supersymmetric, non quantum), I think the limit has to be taken in both sides on the holography. $\endgroup$ – Trimok Jul 28 '13 at 14:00
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In principle, the AdS/CFT correspondence relates a conformal quantum field theory to a quantum theory of gravity (string theory). The key to the solution of all this confusion can be found in taking appropriate limits. It turns out that if you have strong coupling on the string theory side, you have a weakly coupled CFT and vice versa. The weakly coupled limit of string theory is classical (super-gravity), which now corresponds to a strongly coupled conformal quantum field theory. This is one of the main reasons why the correspondence is interesting: it enables one to use perturbative string techniques in order to solve strong coupling field theory problems. There are many efforts to apply this to QCD (which is a strongly coupled field theory), with remarkable success.

But how is this consistent with the assertion that fluid/gravity duality is a classical/classical duality?

Classical on the field theory side in this context does not mean that the underlying theory is not a quantum field theory, it definitely is a strongly coupled QFT. However, in the fluid/gravity duality, the long-wavelength limit is used, which allows one to formulate the problem in terms of the classical Navier-Stokes equation. One can now use the weakly coupled gravity-side to determine parameters for fluid-dynamics.

For a good introduction to the matter at hand, see both http://arxiv.org/pdf/0905.4352v3.pdf and http://arxiv.org/pdf/0712.0689v2.pdf .

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