  It is a well-known property of holographic theories that diffeomorphism
invariance in the bulk space-time implies Weyl invariance of the dual
holographic field theory in the sense that the field theory couples to a
conformal class of background metrics. The usual Fefferman-Graham formalism,
which provides us with a holographic dictionary between the two theories,
breaks explicitly this symmetry by choosing a specific boundary metric and a
corresponding specific metric ansatz in the bulk. In this paper, we show that a
simple extension of the Fefferman-Graham formalism allows us to sidestep this
explicit breaking; one finds that the geometry of the boundary includes an
induced metric and an induced connection on the tangent bundle of the boundary
that is a Weyl connection (rather than the more familiar Levi-Civita connection
uniquely determined by the induced metric). Properly invoking this boundary
geometry has far-reaching consequences: the holographic dictionary extends and
naturally encodes Weyl-covariant geometrical data, and, most importantly, the
Weyl anomaly gains a clearer geometrical interpretation, cohomologically
relating two Weyl-transformed volumes. The boundary theory is enhanced due to
the presence of the Weyl current, which participates with the stress tensor in
the boundary Ward identity.
