By Olivier Biquard

Considering its discovery in 1997 by means of Maldacena, AdS/CFT correspondence has turn into one of many leading topics of curiosity in string idea, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual facet, it presents a duality among a idea 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 development emerged from the disagreement of viewpoints among arithmetic and physics. Written via best specialists and directed at examine 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 large evaluate 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 aspect of the idea. A book of the ecu Mathematical Society. dispensed in the Americas by way of the yankee Mathematical Society.

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

**Example text**

There is another class of closely related metrics for which the fixed point set of ∂ is just a point. These are the AdS Taub–NUT metrics. The line element for these ∂τ metrics has the same form as (5) but the function F (r) is now given by FNUT (r) = Er 2 + (4 − 6E)r 2 + (8E − 8)r + 4 − 3E. (8) Now E is an arbitrary parameter which parameterizes the squashing of the S 3 which is the conformal boundary. The AdS TN and AdS TB have the same asymptotic behavior for k = 1. For |k| > 1, if we identify |k| points on the S 1 fiber of the AdS TN solution, we obtain a space which has the same boundary structure as the AdS TB solution with parameter k.

4) g∈SL(3,Z)/H This expression should be taken with a large grain of salt. We do not really understand how to perform this sum here. 3). The AdS3 string theory should reduce to a Chern–Simons theory at large distances. The calculation of the bulk partition function in this Chern–Simons theory is a state in the space of conformal blocks of the boundary theory and therefore transforms non-trivially under the modular group [32]. 37 Some aspects of the AdS/CFT correspondence On the other hand, the string theory computation in [22] gives a modular invariant partition function.

M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186–225. [31] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998), 105–114. [32] S. Hawking, C. Hunter and D. Page, Nut charge, anti-de Sitter space and entropy, Phys. Rev. D 59 (1999), 044033. [33] S. Hawking and D. Page, Thermodynamics of black holes in Anti-de Sitter space, Comm. Math. Phys. 87 (1983), 577–588. [34] M. Henningson and K.