# Is it possible to build up holography in a closed manifold, i.e., in a manifold with a mathematical boundary?

I was wondering about the AdS/CFT correspondence basics. It is constructed on the idea of conformal compactification, in which a open manifold $M$ is homeomorphic related to a closed one $N$ through a conformal transformation of both metrics. The open manifold acquires a conformal boundary in which the conjecture is stabilized. If one thinks on the Index Theorem, a single issue appears: a pseudo differential operator (e.o.m) on the topological boundary does not have a well defined index, so would not be possible to extend the idea of holography to closed compact spaces?

• now we talk... I also think your suspicion is correct. – user33923 Feb 12 '14 at 6:49