By Olivier Biquard
Considering its discovery in 1997 through Maldacena, AdS/CFT correspondence has turn into one of many best matters of curiosity in string conception, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual part, it offers a duality among a conception of quantum gravity and a box thought. The mathematical counterpart is the relation among Einstein metrics and their conformal limitations. The correspondence has been intensively studied, and many development emerged from the disagreement of viewpoints among arithmetic and physics. Written by way of prime specialists and directed at examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity provides an outline of this crucial region either in theoretical physics and in arithmetic. It includes survey articles giving a huge assessment of the topic and of the most questions, in addition to extra really good articles offering new perception either at the Riemannian facet and at the Lorentzian part of the idea. A e-book of the eu Mathematical Society. allotted in the Americas by means of the yank Mathematical Society.
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Additional info for AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries
Some aspects of the AdS/CFT correspondence 35 simply ZSugra = e−S(Mi ) where S(Mi ) is the classical supergravity action evaluated on the solution Mi . Notice the sum over different manifolds with the same conformal boundary. This plays a crucial role in the study of phase transitions in the boundary CFT. In a certain regime of parameters, generically one of the spaces Mi will dominate the sum on the right-hand side of (1). However, by varying the parameters, which of these Mi ’s dominates may change, leading to phase transitions.
If γ and σ are C ω , then (S, g) is C ω conformally compact. Note that, in contrast to the AH or AdS situation, g(0) and g(3) are freely specifiable on − , subject to smallness conditions. An alternate version of the result should allow one to freely specify g(0) on + and − , provided they are both close to the round metric on S 3 ; this remains to be proved however. 5. This result is an exact analogue of the result of Graham–Lee  on AH Einstein perturbations of the Poincaré metric on the ball B n+1 (since Friedrich’s result predates that of Graham–Lee, the opposite statement is more accurate).
Craig, Dehn filling and asymptotically hyperbolic Einstein manifolds, Thesis, SUNY at Stony Brook, 2004, to appear.  S. deHaro, K. Skenderis and S. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217 (2001), 595–622.  R. Dijkgraaf, J. Maldacena, G. Moore and E. Verlinde, A black hole farey tail, hepth/0005003.  R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero-energy states, J. High Energy Phys.
AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries by Olivier Biquard