Constructing dual gravity theory from general CFT

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AdS/CFT is a little over my head at the moment, particularly the AdS side but I'll ask anyway.

Has there been any work done on starting with a mathematically rigorous conformal field theory defined on the boundary of some space and trying to construct a gravity theory on the interior?

What about the other way? Starting with Type IIB string theory on some space and constructing a conformal field theory from it? Obviously this can be done when the string theory is on $AdS_5\times S^5$ but as far as I'm aware string theory isn't a rigorous mathematical subject yet. To the best of my knowledge CFT is, so that might be a better starting point.

This post imported from StackExchange Physics at 2015-04-13 20:49 (UTC), posted by SE-user ryanp16
asked Apr 13, 2015

Of course people do such a thing! Actually most of the applied AdS/CFT, or for better terminology "gauge/gravity" duality, is based on the following thinking: What is the field theory of the system we are interested about? It is a field theory X. Then by observing closely the symmetries and the behavior of X let's try to construct or to find, if already exists, some "dual" gravitational theory that exhibits the same characteristics but is weakly coupled. For example, I think that the holographic superconductor philosophy is more or less related in this idea. Then, establishing a holographic dictionary is not a difficult task (in most cases) and once having done so people can do calculations for the dual field theory in the bulk. Of course, this dual gravitational theory is not necessary some effective low energy gravitational theory arising from string/M-theory. The AdS/CFT correspondence in that manner is precise string theory in $AdS_5 \times S^5$ is dual to a d=4, $\mathcal{N}=4$ SYM. In the partition function level there is an equal sign actually, not even an approximation, we take the steepest descend approximation (supergravity) because this is what we are able to work with. Therefore, when one starts from a specific QFT and finds the corresponding gravitational theory that suits to this QFT it does not mean this theory arises from strings or M-theory. In the example of the holographic superconductor, I think people have made advances towards this matter (I am really not an expert in such topics so you would have to try to find info on this). An example where people have constructed a theory dual to a specific QFT but do not provide a string background of this theory (because this is not the topic of their investigation of course) is for example this nice paper. As far as the opposite side is concerned, starting from strings/M-theory and constructing QFTs remarkable work has been achieved, mainly because people do NOT try to reproduce, in most cases, some specific QFTs. But even if they do require some specific QFTs their origin is better motivated than the other way around, see for example improved holographic QCD, or the Sakai-Sugimoto model. Finally, as far as how rigorous the above constructions are, I am not sure how to answer but it is true these stuff have not rigorously constructed (unlike CFTs which are well constructed indeed) and I do not think that most physicists worry about this right now. Normal QFT path integral has also some some problems with its rigorous formulation but using it we obtain SM results in the experiments if this has any importance to the rigorous math question of yours.
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