Weyl connections and their role in holography

Abstract

It is a well known property of holographic theories that diffeomorphism invariance in the bulk spacetime 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.

Gauges in three-dimensional gravity and holographic fluids

“…Indeed, the remaining uu, uφ and φφ equations are the uu, * u * u and * uu equations in the anti-de Sitter triad, or the µ µ µµ µ µ, * µ µ µ * µ µ µ and * µ µ µµ µ µ in the flat counterpart. 18 Using the expressions for the expansions displayed in footnotes 5 and 12, we obtain Θ * k = θ * = − 1 Γ ∂ φ ln γ = 2e −ϕ ∂ φ β0 and Θ = θ = γ ∂u ln Γ + v φ ∂ φ ln Γ + ∂ φ v φ = e −2β 0 (∂uϕ + U0∂ φ ϕ + ∂ φ U0).…”

“…The fluid/gravity geometric and kinematic data Γ, v φ and γ are related to the partly on-shell Bondi functions ϕ, U 0 and β 0 as follows: 18 Γ = e ϕ , v φ = U 0 , γ = e −2β 0 .…”

“…Its relation to the fluid/gravity gauge is also discussed there, and amounts to simply switching off one of the six functions 1 Extend the Fefferman-Graham gauge so that Weyl covariance be manifest has been achieved only recently in ref. [18]. 2 In ref.…”

Fefferman--Graham and Bondi Gauges in the Fluid/Gravity Correspondence

“…Here and in the following, "locally" means that the boundary-metric data are unspecified 3. See[45] for an explicit Weyl-covariant enhancement of this gauge.…”

Weyl Prior and Bayesian Statistics

“…Einstein soon pointed out that Weyl’s theory predicted substantial broadening of the characteristic length of atoms, which is contradictory to the well-observed thin atomic spectra. Even though Weyl geometry failed the unification of gravity and electromagnetism, which is still an open problem, Weyl geometry has found applications in possible generalization of general relativity [ 5 ] and the differential-geometric study of defects in continuum mechanics [ 6 ]. Weyl geometry is kept in mathematics.…”