While editing the tag wiki for ads-cft, 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?
Best Answer
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 https://arxiv.org/abs/0905.4352 and https://arxiv.org/abs/0712.0689.