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I would like to inquire about references for the AdS/CFT correspondence in dimensions 3 and 2, namely between $AdS^3$ (and gravity there) and its 2-dimensional boundary at infinity (and CFT there).

Here are some constraints: if possible, I would like references that are Mathematician-friendly (or at least readable by someone who is familiar with the basics of QFT but without much depth), and I am mostly interested in the Riemannian signature case actually (or what Physicists call the Euclidean case). So actually, I am interested in a correspondence between gravity on hyperbolic 3-space $H^3$ and CFT on its sphere at infinity. I would be OK with something "formal" and not directly applicable to the "real world", so to speak. If someone knows of some references that could be useful, and not necessarily satisfying all my constraints, then please share them as well.

Edit: I thank everyone who has provided some references. I would like to make a little more precise what I am looking for. I am interested in references that explain what corresponds on the boundary to a configuration of $n$ electrons in the bulk. I am mostly interested in the special case where the bulk is 3-dimensional. I am particularly interested in references containing information and formulae on the relevant partition functions. I apologize for modifying my original post in this edit. I hope this makes it a little more precise what I am looking for.

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Maldacena proved in 1997, that there is a weak-strong duality between quantum gravity in D+1 dimensions and Yang-Mills theory formulated on and Anti De Sitter space in D dimensions. Seminal articles (must read):

https://arxiv.org/abs/hep-th/9812012

http://lanl.arxiv.org/abs/hep-th/9711200

http://lanl.arxiv.org/abs/hep-th/9804085

I hope this helps!

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  • $\begingroup$ Thank you so much. You have a valid point. I am interested though in a special case which should be easier to prove than the general case. Thank you for the references though. I should definitely read those! $\endgroup$ – Malkoun May 28 '17 at 4:43
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I found literature on quantum field theory on hyperbolic space, in particular $H3$. I copied a few sentences from an articleHyperbolic spaces in String theory and M theory so you can decide if you want to read the rest. The quote is

"4 Newton’s law on H3 Having described a number of string compactifications containing hyperbolic spaces, a question of interest (in view of the recent results of [23] for Newton’s law in certain noncompact spaces) is whether such compactifications may be compatible with the observed Newton’s law. In the case of compactification on a compact space X, there exist massive Kaluza-Klein modes of the four-dimensional graviton associated with the eigenvalues of 7 the Laplace operator on X. In the static limit, these modes contribute to the Newton’s law giving, in addition to the standard 1/r behavior of the massless graviton, Yukawa type corrections." End of quote.

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