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.
Highlights
There is an important fact about the asymptotic AdS geometry: the conformal boundary of a (d þ 1)dimensional asymptotically locally anti–de Sitter (AlAdS) spacetime carries not a metric but a conformal class of metrics; i.e., the boundary enjoys Weyl symmetry
Our goal in this paper is to find the holographic Weyl anomaly in higher dimensions using the advantages of the WFG gauge, and organize the results in a form that manifests its general structure
Our results reveal some interesting clues about the general form of the holographic Weyl anomaly in any dimension
Summary
There is an important fact about the asymptotic AdS geometry: the conformal boundary of a (d þ 1)dimensional asymptotically locally anti–de Sitter (AlAdS) spacetime carries not a metric but a conformal class of metrics; i.e., the boundary enjoys Weyl symmetry. In the FG gauge, after going through the holographic renormalization procedure by adding counterterms to cancel the divergence extracted by the regulator, one finds that the holographic Weyl anomaly in an even dimension corresponds to the logarithmic term in the bulk volume expansion In mathematical literature this is referred to as the Q curvature [29,30,31,32] (see [33] for a short review), which has been studied by means of obstruction tensors and extended obstruction tensors in [7,8]. Going into the WFG gauge, it was shown in [5] using dimensional regularization that the Weyl anomaly in the 2k dimension can be extracted directly from the variation of the pole term at the Oðz2k−dÞ order of the “bare” on-shell action under the d → 2k− limit This is the method we will use for computing the Weyl anomaly in this work. VII we summarize our results and point out possible directions for future studies
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