Weyl Cohomology and the Effective Action for Conformal Anomalies

TL;DR

The paper develops a general, cohomological framework for conformal (trace) anomalies in even dimensions by defining a Weyl coboundary operator and identifying non-trivial Weyl cocycles with Weyl-variant counterterms in dimensional regularization that become Weyl invariant only at the physical dimension $d=2k$. This yields a finite, non-local Wess-Zumino action $\Gamma_{WZ}[g;\sigma]$ whose variation reproduces the anomaly and is unique up to Weyl-invariant and trivial cocycles; the non-trivial cocycles correspond to infrared relevant, long-distance physics in gravity, while trivial ones are UV-centric. In $d=2$ and $d=4$ the construction reproduces Polyakov’s action and its four-dimensional analogue, with an explicit non-local form $S_{anom}$ built from $F_4$ and $E_4$ and an auxiliary scalar $\varphi$; the associated energy–momentum tensors explain IR dynamics, including the appearance of the locally conserved $^{(3)}H_{ab}$ in conformally flat spacetimes. The Fefferman–Graham embedding provides conformal invariants in arbitrary even dimensions, and dimensional regularization links the cohomology classification to holographic ideas (AdS/CFT and de Sitter/CFT), showing how infrared anomaly terms modify low-energy gravity and clarifying the structure of the low-energy effective action.

Abstract

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, and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with respect to this group. Non-trivial cocycles of the Weyl group arise from local functionals that are Weyl invariant in and only in the physical even integer dimension. In the physical dimension the non-trivial cocycles generate covariant non-local action functionals characterized by sensitivity to global Weyl rescalings. The non-local 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 $d=4$ non-local effective action yields two new conserved geometric stress tensors with local traces. 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 behavior of correlators at the asymptotic infinity of either anti-de Sitter or de Sitter spacetimes follows, i.e. AdS$_{d+1}$ or deS$_{d+1}$/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 imply that the classical Einstein theory is modified at large distances.