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

Let $M$ be an $\left(n+1\right)$-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. $\left(M,g\right)$ 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 $\left(N,c\right)$ should be a sum of terms corresponding to the string theory on the $\left({M}_{i},{g}_{i}\right)$, where $\partial {M}_{i}=N$ and the ${g}_{i}$ are Einstein conformally compact metrics on ${M}_{i}$ inducing $c$ on $N$.

The CFT on $N$ makes sense mostly when $\left(N,c\right)$ 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, $\partial M=N$ and $g$ is an Einstein, conformally compact metric on $M$ inducing $c$ on $N$, then ${H}_{n}\left(M,Z\right)=0$.

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 ${H}_{n}\left(M,Z\right)$.

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:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C20 Global Riemannian geometry, including pinching 53C80 Applications of global differential geometry to physics 81T30 String and superstring theories 83E30 String and superstring theories