Obstruction Tensors in Weyl Geometry and Holographic Weyl Anomaly

Abstract

Recently a generalization of the Fefferman-Graham gauge for asymptotically locally AdS spacetimes, called the Weyl-Fefferman-Graham (WFG) gauge, has been proposed. It was shown that the WFG gauge induces a Weyl geometry on the conformal boundary. The Weyl geometry consists of a metric and a Weyl connection. Thus, this is a useful setting for studying dual field theories with background Weyl symmetry. Working in the WFG formalism, we find the generalization of obstruction tensors, which are Weyl-covariant tensors that appear as poles in the Fefferman-Graham expansion of the bulk metric for even boundary dimensions. We see that these Weyl-obstruction tensors can be used as building blocks for the Weyl anomaly of the dual field theory. We then compute the Weyl anomaly for $6d$ and $8d$ field theories in the Weyl-Fefferman-Graham formalism, and find that the contribution from the Weyl structure in the bulk appears as cohomologically trivial modifications. Expressed in terms of the Weyl-Schouten tensor and extended Weyl-obstruction tensors, the results of the holographic Weyl anomaly up to $8d$ also reveal hints on its expression in any dimension.

37 pages; v3: minor typos fixed

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• Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions
• The Ambient Obstruction Tensor and Q-Curvature
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• Weyl-Ambient Geometries
• 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
• Holographic Weyl Anomaly and Kounterterms in AdS gravity