Weyl cohomology and the effective action for conformal anomalies

We present a general method of deriving the effective action for conformal anomalies in any even dimension, which satisfies the Wess-Zumino consistency condition by construction. The method relies on defining the coboundary operator of the local Weyl group, $g_{\mathrm{ab}}\to \exp (2\mathrm{}\sigma {)g}_{\mathrm{ab}},$ and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with respect to this group. Nontrivial cocycles of the Weyl group arise from local functionals that are Weyl invariant in and only in the physical even integer dimension $d=2k.\mathrm{}$ In the physical dimension the nontrivial cocycles generate covariant nonlocal action functionals characterized by sensitivity to global Weyl rescalings. The nonlocal action so obtained is unique up to the addition of trivial cocycles and Weyl invariant terms, both of which are insensitive to global Weyl rescalings. These distinct behaviors under rigid dilations can be used to distinguish between infrared relevant and irrelevant operators in a generally covariant manner. Variation of the $d=4\mathrm{}$ nonlocal effective action yields two new conserved geometric stress tensors with local traces equal to the square of the Weyl tensor and the Gauss-Bonnet-Euler density, respectively. The second of these conserved tensors becomes ${}^{\mathrm{}\left(3\right)\mathrm{}}{H}_{\mathrm{ab}}$ in conformally flat spaces, exposing the previously unsuspected origin of this tensor. The method may be extended to any even dimension by making use of the general construction of conformal invariants given by Fefferman and Graham. As a corollary, conformal field theory (CFT) behavior of correlators at the asymptotic infinity of either anti–de Sitter or de Sitter spacetimes follows, i.e., ${\mathrm{AdS}}_{d+1\mathrm{}}-$ or ${\mathrm{deS}}_{d+1\mathrm{}}-{\mathrm{CFT}}_d$ correspondence. The same construction naturally selects all infrared relevant terms (and only those terms) in the low energy effective action of gravity in any even integer dimension. The infrared relevant terms arising from the known anomalies in $d=4\mathrm{}$ imply that the classical Einstein theory is modified at large distances.