By Olivier Biquard

Because its discovery in 1997 by way of Maldacena, AdS/CFT correspondence has develop into one of many top topics of curiosity in string thought, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual part, it presents a duality among a conception of quantum gravity and a box idea. The mathematical counterpart is the relation among Einstein metrics and their conformal obstacles. The correspondence has been intensively studied, and many development emerged from the war of words of viewpoints among arithmetic and physics. Written by means of top specialists and directed at examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity supplies an outline of this crucial zone 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 offering new perception either at the Riemannian part and at the Lorentzian facet of the idea. A booklet of the eu Mathematical Society. disbursed in the Americas by way of the yankee Mathematical Society.

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

**Example text**

7) Geometric aspects of the AdS/CFT correspondence 23 Formally, this is obtained from g = g − by the replacement τ → iτ , ρ → iρ, interchanging a spatial and time direction. This gives an extension of the metric g into a region M ext exterior to the light cone H , defined for all τ ∈ (0, ∞), ρ ∈ [0, ε), for some ε > 0. The metric gext is C ω where ρ = 0, but is only C 3/2 up to H where ρ = 0. The metric gext is also a Lorentzian cone metric, now however with a spacelike self-similarity in τ in place of the previous time-like self-similarity.

3 Let I0 be a connected component of I of M . e. J + (I0 ) (I \ I0 ) = ∅. In the Euclidean case, for asymptotically AdS Einstein space-times, the puzzle is avoided due to a theorem by Witten and Yau [62], basically stating that if one of the boundary components has R > 0, then the boundary is connected. This theorem was later generalized by Cai and Galloway [17] to cases where the boundary has zero scalar curvature. 5 If N has a component of nonnegative curvature, then the following holds: (i) N is connected; (ii) If M is 2 One example for this is the case of Schwarzschild AdS black holes, and in particular the BTZ black hole.

Theor. Math. Phys. 2 (1998), 253–291. [48] E. -T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), 1635–1655. nl Abstract. This is a very brief review of some aspects of the AdS/CFT correspondence with an emphasis on the role of the topology of the boundary and the meaning of the sum over bulk geometries. 1 Introduction Since its incarnation in 1997 [47], [30], [59] (for a review see [1]), the AdS/CFT correspondence has been one of the prime subjects of interest in string theory.