  Field theories with weakly coupled holographic duals, such as large N gauge
theories, have a natural separation of their operators into `single-trace
operators' (dual to single-particle states) and `multi-trace operators' (dual
to multi-particle states). There are examples of large N gauge theories where
the beta functions of single-trace coupling constants all vanish, but marginal
multi-trace coupling constants have non-vanishing beta functions that spoil
conformal invariance (even when all multi-trace coupling constants vanish). The
holographic dual of such theories should be a classical solution in anti-de
Sitter space, in which the boundary conditions that correspond to the
multi-trace coupling constants depend on the cutoff scale, in a way that spoils
conformal invariance. We argue that this is realized through specific bulk
coupling constants that lead to a running of the multi-trace coupling
constants. This fills a missing entry in the holographic dictionary.
