The AdS/CFT correspondence postulates a duality between string theory of gravity and a CFT on an AdS background. This duality is employed in some condensed matter systems. I was wondering why it is applicable here when the condensed matter system in a laboratory is described by a quantum field theory that doesn't reside in AdS space - our space isn't AdS. Keep in mind I have studied general relativity and quantum field theory but not string theory.
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1$\begingroup$ AdS/CFT correspondance is a duality between a gauge theory (a QFT) and a gravitational theory. A condensed matter system is then on the "gauge" side. $\endgroup$– KosmFeb 13, 2019 at 9:35
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1$\begingroup$ Usually the idea is to convert a strongly-coupled condensed-matter system into a CFT (on Minkowski spacetime), which can then be translated into an AdS gravitational system. $\endgroup$– probably_someoneFeb 13, 2019 at 10:34
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You mixed it up.
CFT lives in the flat Minkowski spacetime. It is the gravitional side of the equivalence that lives in the AdS spacetime. That's why in the name you have AdS and CFT separated by slash - they denote two different sides of the correspondence.
E.g. the nost famous case is the equivalence of the $\mathcal{N}=4$ supersymmetric Yang-Mills theory in the 4-dimensional Minkowski spacetime (which is CFT) with the Type IIb string theory in $AdS_5\times S_5$.
One of the most basic signs of the equivalence is that the conformal group of the $d$-dimensional Minkowski spacetime $SO(2,d)$ is exactly the isometry group of the $d+1$-dimensional AdS.
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$\begingroup$ Thanks. In general does the CFT strictly have be in Minkowski space for the AdS/CFT correspondence to be applicable or is it possible for there to be some curvature of space-time? $\endgroup$– Matrix23Feb 15, 2019 at 0:21
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$\begingroup$ @IanDsouza I'm probably not the best personb to ask. When you consider CFT in Minkowski spacetime it is dual not to the full AdS spacetime but to the so called Poincare patch. It is generally understood that the full global AdS is dual to the CFT on a cylinder. I've seen some attempts at similar construction of the CFT on dS as a certain slicing of the AdS spacetime though not sure how well-motivated they are. Also, the way I see it, the dual for finite slice of AdS should not be completely decoupled from gravity though it seems that this is usually neglected in applications. $\endgroup$– OONFeb 18, 2019 at 14:19
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$\begingroup$ @IanDsouza The last thing is usually not neglected when people talk about the holographic interpretation of the Randall-Sundrum scenarios. Of course then your QFT coupled to gravity can only be understood as an effective field theory $\endgroup$– OONFeb 18, 2019 at 14:20