By Olivier Biquard

Because its discovery in 1997 through Maldacena, AdS/CFT correspondence has develop into one of many major topics of curiosity in string concept, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual part, it presents a duality among a thought of quantum gravity and a box conception. The mathematical counterpart is the relation among Einstein metrics and their conformal barriers. The correspondence has been intensively studied, and many development emerged from the war of words of viewpoints among arithmetic and physics. Written by way of top specialists and directed at examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity offers an summary of this crucial quarter either in theoretical physics and in arithmetic. It includes survey articles giving a large assessment of the topic and of the most questions, in addition to extra really expert articles offering new perception either at the Riemannian facet and at the Lorentzian part of the idea. A booklet of the eu Mathematical Society. dispensed in the Americas through the yank Mathematical Society.

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

**Example text**

We examine this issue on a particular family of examples. 6. Let g − be the AdS–Taub–NUT metric on B 4 , cf. 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 − .

The exact map between the bulk and the boundary theory is still somewhat mysterious. For more details, see [8]. 6 The field is constant in the sense that the field strength of the U(1) connection is proportional to the AdS2 volume form. e. wormholes), we must look at configurations which do not satisfy the conditions of the Witten–Yau theorem. One possibility is to look at spaces which have two boundaries, each being a Riemann surface of genus g ≥ 2, g [49]: ds 2 = dρ 2 + cosh2 ρ ds 2 g . (27) The boundaries have constant negative curvature, so the Witten–Yau theorem does not apply here, but unlike in higher dimensions, the two-dimensional field theories on these Riemann surfaces are well-defined and stable.

Phys. 208 (1999), 413–428. [13] O. Biquard, Métriques d’Einstein asymptotiquement symmétriques, Astérisque 265 (2000). [14] O. Biquard, Métriques autoduales sur la boule, Invent. Math. 148 (2002), 545–607. [15] D. Brill and F. Flaherty, Maximizing properties of extremal surfaces in General Relativity, Ann. Inst. H. Poincaré Sect. ) 28 (1978), 335–347. [16] M. Cai and G. Galloway, Boundaries of zero scalar curvature in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), 1769–1783. [17] D.