  We discuss the general question of which conformal field theories have dual
descriptions in terms of quantum gravity theories on anti-de Sitter space. We
analyze in detail the case of a deformed product of n conformal field theories
(each of which has a gravity dual), and we claim that the dual description of
this is by a quantum gravity theory on a union of n anti-de Sitter spaces,
connected at their boundary (by correlations between their boundary
conditions). On this union of spaces, (n-1) linear combinations of gravitons
obtain a mass, and we compute this mass both from the field theory and from the
gravity sides of the correspondence, finding the same result in both
computations. This is the first example in which a graviton mass in the bulk of
anti-de Sitter space arises continuously by varying parameters. The analysis of
these deformed product theories leads us to suggest that field theories may be
generally classified by a "connectivity index", corresponding to the number of
components (connected at the boundary) in the space-time of the dual
gravitational background. In the field theory this index roughly counts the
number of independent gauge groups, but we do not have a precise general
formula for the index.
