Weyl Connections and their Role in Holography

TL;DR

This work extends holography by incorporating a boundary Weyl connection through a Weyl–Fefferman–Graham gauge, so the boundary geometry becomes Weyl-covariant with an accompanying Weyl current. The authors show that bulk diffeomorphisms corresponding to Weyl rescalings act naturally on boundary data, inducing a Weyl connection on the boundary and rendering all subleading bulk terms Weyl-covariant. The holographic dictionary is reformulated to source both the boundary stress tensor and the Weyl current, leading to a Weyl Ward identity that links the trace of T^{μν} to the divergence of J^μ, and to a Weyl-anomaly expressed in Weyl-covariant terms with explicit results in d=2 and d=4. This framework provides a geometric interpretation of the Weyl anomaly and sets the stage for deeper field-theory analyses in backgrounds (γ^(0), a^(0)) beyond the standard metric-only sourcing.

Abstract

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.