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Connectedness of the boundary in the AdS/CFT correspondence. (English) Zbl 0978.53085

Let M be an (n+1)-dimensional manifold with non-empty boundary N. A metric g on M is conformally compact if it can be written as h/r 2 , where h is a compact metric on M, and r is a function on M with a first-order zero on the boundary. (M,g) is then complete, and its boundary has a well-defined conformal structure.

The AdS-CFT correspondence is an important conjecture which states that, maybe after taking the tensor product of M with a compact manifold, the conformal field theory (CFT) on N should be related to the string theory in M. More precisely, if c is a conformal class on N, the partition function of the CFT on (N,c) should be a sum of terms corresponding to the string theory on the (M i ,g i ), where M i =N and the g i are Einstein conformally compact metrics on M i inducing c on N.

The CFT on N makes sense mostly when (N,c) has positive scalar curvature, in the sense that there is a metric on N conformal to c which has positive scalar curvature. The main result of the paper is that if, in addition, M=N and g is an Einstein, conformally compact metric on M inducing c on N, then H n (M,Z)=0.

The authors point out that this implies that N is connected, and also that M contains no “wormhole”; it therefore removes some puzzling questions concerning the AdS-CFT correspondence.

To prove the result, the authors introduce the “brane action” L on embedded hypersurfaces of M, defined as the area minus n times a primitive of the volume form. Then they prove using a local computation that L has no minimum, while the behavior of the function at infinity shows that there is one in each non-zero class of H n (M,Z).

Since this paper appeared, several alternate proofs of its main result have been provided, in particular by M. T. Anderson [Adv. Math. 179, No.2, 205-249 (2003; Zbl 1048.53032); see also math.DG/0104171], by M. Cai and G. J. Galloway [Adv. Theor. Math. Phys. 3, No. 6, 1769-1783 (2000; Zbl 0978.53084)] and by X. Wang (preprint).


MSC:
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C20Global Riemannian geometry, including pinching
53C80Applications of global differential geometry to physics
81T30String and superstring theories
83E30String and superstring theories