I have a question regarding AdS/CFT and what it teaches about our world. AdS/CFT if often treated as our best tool to explore quantum gravity, and people have worked very hard in trying to understand the dictionary on how gravity emerges from a strongly coupled, large $N$, field theory, in particular, how spacetime emerges. This is treated as a very interesting area because one is able to explore quantum gravity by using the well-known methods of field theory.
However, all I have seen on AdS/CFT so far deals with gravity on asymptotically AdS spacetime. This asymptotics is $\it{crucial}$ for the correspondence to work, for many reasons. I've also seen that approaches to dS or flat space holography are in a very primitive stage, and their consistency is by no means assured.
Why then are people are so interested in AdS/CFT from the point of view of a definition of quantum gravity? Suppose you are finally able to explain all entries of the AdS/CFT dictionary. Is there any, even if very rough, way in which to extrapolate this to flat space or dS space? Is there at least any small indication that this could be possible?
Answer
I think most of the questions are a little bit subjective and often the answer is currently not known. So I will just give my opinion.
Why then are people are so interested in AdS/CFT from the point of view of a definition of quantum gravity?
Since quantum gravity is not very well understood in general I would say any tool that helps you to analyze it is very much welcome. One might hope that some of the aspects learned via the duality are true irrespective of the value of the cosmological constant. Of course, some aspects change drastically e.g., the asymptotic structure.
Is there any, even if very rough, way in which to extrapolate this to flat space or dS space? Is there at least any small indication that this could be possible?
This is part of current research (see e.g., https://inspirehep.net/search?ln=en&p=t+flat+holography and https://inspirehep.net/search?ln=en&ln=en&p=t+de+sitter+holography), but some ideas are around e.g., for flat Leonard Susskind - Holography in the flat space limit and on a more rigorous level Dappiaggi, Moretti, Pinamonti - Rigorous steps towards holography in asymptotically flat spacetimes and for dS e.g.,
Balasubramanian, de Boer, Minic - Notes on de Sitter space and holography and Larsen, van der Schaar, Leigh - De Sitter holography and the cosmic microwave background.
Let me maybe add in general, that the holographic principle is in principle independent of AdS. The AdS/CFT correspondence is just a realization of it. See e.g., one of the papers that started it 't Hooft - Dimensional reduction in quantum gravity Gerard where AdS is nowhere mentioned. So one might hope that it holds beyond AdS.
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