Weyl-Ambient Geometries

Abstract

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal ${\mathbb{R}}_{+}$-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.

42 pages, 1 figure; v4: references added

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• Symmetries and Charges in Weyl-Fefferman-Graham Gauge
• Radiation in Fluid/Gravity and the Flat Limit
• BRST Cohomology is Lie Algebroid Cohomology
• Field-dependent diffeomorphisms and the transformation of surface charges between gauges