  We discuss the constraints that a conformal field theory should enjoy to
admit exactly marginal deformations, i.e. to be part of a conformal manifold.
In particular, using tools from conformal perturbation theory, we derive a sum
rule from which one can extract restrictions on the spectrum of low spin
operators and on the behavior of OPE coefficients involving nearly marginal
operators. We then consider conformal field theories admitting a gravity dual
description, and as such a large-$N$ expansion. We discuss the relation between
conformal perturbation theory and loop expansion in the bulk, and show how such
connection could help in the search for conformal manifolds beyond the planar
limit. Our results do not rely on supersymmetry, and therefore apply also
outside the realm of superconformal field theories.
