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I have a good picture of AdS/CFT correspondence when the AdS space is given in terms of Global coordinates. In global coordinates, the AdS space is just a cylinder (up to a conformal factor) and gravity theory that is living within that cylinder has an equivalent description in terms of a CFT that can be considered to be living on the boundary of the cylinder. But I can also work in Poincare coordinates. So the CFT is living on the Poincare patch.

a) Now what is the relation between the CFT that is on the Poincare patch to the CFT that is on the complete boundary of the cylinder ?

b) Since Poincare patch is just a sub-region of the boundary of the cylinder, is it even reasonable to expect a consistent CFT will live on the Poincare patch, given a CFT on the complete space? Wouldn't the excitations from other parts of space-time make it impossible to consider a field theory defined just on a sub-region?

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Lüscher and Mack showed in this paper that if you have CFT correlation functions defined on a Poincare patch, you can analytically continue these correlation functions to the whole Lorentzian cylinder. From the point of view of the Lorentzian cylinder the reason you can consistently think about a theory as living on the Poincare patch is that there is a conformal generator which preserves it and thus can be taken to be the Poincare patch Hamiltonian. The spatial slice on Poincare patch is the same as on the cylinder and so the Hilbert space is already the same in both cases.

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  • $\begingroup$ @Ganesh regarding your questions about excitations from the other parts of Lorentzian cylinder, from the point of view of the Poincare patch, they just modify the initial or the final state. $\endgroup$ – Peter Kravchuk Nov 20 '17 at 4:12

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