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.