By Olivier Biquard

ISBN-10: 3037190132

ISBN-13: 9783037190135

Due to the fact that its discovery in 1997 by means of Maldacena, AdS/CFT correspondence has develop into one of many major topics of curiosity in string thought, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual facet, it offers a duality among a thought of quantum gravity and a box conception. 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 through major specialists and directed at examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity offers an outline of this significant region either in theoretical physics and in arithmetic. It comprises survey articles giving a large evaluation of the topic and of the most questions, in addition to extra really good articles delivering new perception either at the Riemannian aspect and at the Lorentzian facet of the idea. A book of the ecu Mathematical Society. dispensed in the Americas by means of the yank Mathematical Society.

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**Additional resources for AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries **

**Sample text**

In the correspondence between AH 24 Michael T. 12) where + occurs if k ≡ 0, 1 (mod 4), while − occurs if k ≡ 2, 3 (mod 4). 19) for the renormalized action and its variation also have analogues for dS space-times. 10) which is asymptotically simple, in that (S, g) has a smooth past and future conformal infinity ( − , γ − ) and ( + , γ + ). In particular, S is geodesically complete and globally hyperbolic with compact Cauchy surface , a 3-manifold diffeomorphic to − and + . 11) is usually replaced by its negative for Lorentzian metrics.

22) where E ∈ (0, ∞) is any constant, r ≥ 1, and F (r) = Er 4 + (4 − 6E)r 2 + (8E − 8)r + 4 − 3E. 23) The length of the S 1 parametrized by θ1 is 2π . This metric is self-dual Einstein and has conformal infinity γ − given by the Berger (or squashed) sphere with S 1 fibers of length β = 2π E 1/2 over S 2 (1). Clearly γ − is C ω , as is the geodesic compactification with boundary metric γ − . 5). When E = 1, g − is the Poincaré metric. The deSitter continuation of g − is the dS–Taub–NUT metric on R × S 3 , cf.

5. This result is an exact analogue of the result of Graham–Lee [30] 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). It would be very interesting if a higher dimensional analogue of Friedrich’s result could be proved, as in the Graham–Lee theorem. We may now apply this result to the “initial” AH Einstein metric (M, g), g = g − . Thus, on M = B 4 , let g − be an AH Einstein metric with C ω boundary metric γ − close to the round metric γ+1 on S 3 , (so g − is close to the Poincaré metric on B 4 ).

### AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries by Olivier Biquard

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