By Olivier Biquard
In view that its discovery in 1997 via Maldacena, AdS/CFT correspondence has turn into one of many best topics of curiosity in string idea, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual aspect, it offers a duality among a conception of quantum gravity and a box idea. The mathematical counterpart is the relation among Einstein metrics and their conformal limitations. The correspondence has been intensively studied, and many growth emerged from the war of words of viewpoints among arithmetic and physics. Written through best specialists and directed at learn mathematicians and theoretical physicists in addition to graduate scholars, this quantity supplies an outline of this crucial quarter either in theoretical physics and in arithmetic. It comprises survey articles giving a wide evaluation of the topic and of the most questions, in addition to extra really good articles supplying new perception either at the Riemannian facet and at the Lorentzian facet of the speculation. A e-book of the ecu Mathematical Society. allotted in the Americas through the yankee Mathematical Society.
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Extra info for AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries
For more details, see . 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 : 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.
7) then give ¯ρ ρ ¯ρ ρ + (R − n(n + 1))/ρ 2 ≥ −2n ¯ρ ρ . + (Ricg −ng)(T , T )/ρ 2 . 17) where again φ = − ¯ ρ/ρ. 8) also holds, and hence Rγ < 0 implies φ(0) < 0. 16). 3 to the situation where Rγ = 0. 1, (S, g) has + = ∅ and no time-like geodesic is futurecomplete unless (S, g) is isometric to g = −dt 2 + e2t gN n , where gN n is Ricci-flat, cf.  for further details. 15) cannot be geodesically complete if Rγ < 0, and have at most one component of conformal infinity. This exhibits the role of the hypothesis Rγ > 0 in a more drastic way than the AH case.
8), where T = ∇ρ is the future-directed unit vector. We summarize some of this discussion in the following result. 1. 7). 8). This correspondence thus gives a rigorous form of “Wick rotation” between these types of Einstein metrics. The Fefferman–Graham expansion holds for Einstein metrics of any signature (again as observed in ). 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.
AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries by Olivier Biquard