Holographic Trace Anomaly and Cocycle of Weyl Group

TL;DR

This work analyzes the 1-cocycle of the Weyl group, i.e., the holographic trace anomaly, by examining how the logarithmically divergent bulk AdS/CFT effective action responds to finite diffeomorphisms that enact boundary Weyl transformations. Using the Henningson-Skenderis holographic framework and Fefferman-Graham expansion, the authors extract the bulk divergences ${\bf a}_{(2)}$ and ${\bf a}_{(4)}$ and show that their Weyl-transformed bulk action reproduces the known 2D and 4D cocycles, matching the standard expressions ${\cal A}_2 = -\frac{c}{24\pi}R$ and ${\cal A}_4 = \alpha E_4 + \beta I_4$, with cocycles $S^{d=2}(\sigma,g)$ and $S^{d=4}(\sigma,g)$ agreeing with cohomological results. The normalization is tied to the central charge via $1/(2 k_3^2) = c/(24\pi)$, reinforcing the consistency between holography and CFT anomaly structures. The paper suggests extensions to $d=6$ and discusses implications for renormalization of anomaly coefficients across weak/strong coupling, including the $({\cal N}=4)$ SYM and (2,0) theories.

Abstract

The behavior of the divergent part of the bulk AdS/CFT effective action is considered with respect to the special finite diffeomorphism transformations acting on the boundary as a Weyl transformation of the boundary metric. The resulting 1-cocycle of the Weyl group is in full agreement with the 1-cocycle of the Weyl group obtained from the cohomological consideration of the effective action of the corresponding CFT.