Connectedness Of The Boundary In The AdS/CFT Correspondence

Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary N of positive scalar curvature. We show that under these conditions, H n (M ; Z) = 0 and in particular N must be connected. These results resolve some puzzles concerning the Suppose that M is a complete Einstein manifold of negative curvature and dimension n + 1 and that the conformal boundary of M , in the sense of Penrose [1], is an n-manifold N. This means that M is the interior of an n + 1-dimensional manifold-with-boundary M , whose boundary is N , and that the metric g of M can be written near the boundary as g = 1 t 2 dt 2 + g ij (x, t)dx i dx j , (1.1) where t is a smooth function with a first order zero on ∂M , and positive on M , and g ij (x, t)dx i dx j is an t-dependent family of metrics on M. Thus, t ≥ 0 on M and t = 0 on N. In this situation, g 0 = g(x, 0) is a metric on N. If t is replaced by a different function with a first order zero on ∂M , say t ′ = e ω t, then g 0 undergoes a conformal transformation g 0 → g ′ 0 = e 2ω g 0 , so N actually has a natural conformal structure but not a natural metric. If, in the conformal class of metrics on N , there is a representative with positive (or zero, or negative) scalar curvature, then we say that N has positive (or zero, or negative) scalar curvature. There is a correspondence between conformal field theory on N and quantum gravity, or string theory, on M [2-4]. To be more precise, the correspondence asserts (see [4], section 3) that to do conformal field theory on N with a given conformal structure g 0 on N , one must sum over contributions of all possible n + 1-dimensional Einstein manifolds M with conformal boundary N and induced conformal structure g 0. Actually, the full correspondence involves a number of additional details that we will omit in the present paper. For example, one usually must consider not n + 1-dimensional Einstein manifolds M , but manifolds …