By Olivier Biquard

Because its discovery in 1997 by means of Maldacena, AdS/CFT correspondence has develop 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 aspect, 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 obstacles. The correspondence has been intensively studied, and many growth emerged from the disagreement of viewpoints among arithmetic and physics. Written by way of prime specialists and directed at examine mathematicians and theoretical physicists in addition to graduate scholars, this quantity provides an outline of this significant zone either in theoretical physics and in arithmetic. It includes survey articles giving a extensive assessment of the topic and of the most questions, in addition to extra really good articles delivering new perception either at the Riemannian facet and at the Lorentzian aspect of the idea. A ebook of the ecu Mathematical Society. dispensed in the Americas through the yankee Mathematical Society.

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8 (2001), 171–188. [4] M. Anderson, Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds, Adv. Math. 179 (2003), 205–249. [5] M. DG/0105243. [6] M. Anderson, Some results on the structure of conformally compact Einstein metrics, preprint, Feb. DG/0402198. [7] M. DG/0303260. [8] M . Anderson, Remarks on evolution of spacetimes in 3 + 1 and 4 + 1 dimensions, Classical Quantum Gravity 18 (2001), 5199–5209. [9] M. Anderson, S. Carlip, J. Ratcliffe, S. Surya and S.

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.

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. It provides a duality between a theory with quantum gravity in d dimensions and a field theory in d − 1 dimensions.