Trace Anomalies and Cocycles of Weyl and Diffeomorphisms Groups

TL;DR

The paper investigates trace (Weyl) anomalies in even dimensions through the Wess-Zumino consistency condition, arguing that the anomaly structure comprises Euler densities, Weyl-invariant curvature polynomials, and coboundaries from local counterterms. It constructs finite Weyl cocycles in $d=4$ and $d=6$, revealing a deep link between nontrivial cocycles and conformal-invariant operators of order $d$ (with a second-order $\Delta_4$ in $d=4$ and a (not unique) sixth-order $\Delta_6$ in $d=6$); it also shows how to reduce these cocycles to second order by adding trivial counterterms. Furthermore, the authors map Weyl cocycles to diffeomorphism cocycles, yielding explicit expressions for $Diff(4)$ and $Diff(6)$ via a Weyl-invariant regularization, thus providing a unified framework connecting trace anomalies, cocycles, and diffeomorphism anomalies. The work suggests a general pattern for higher dimensions and offers a systematic route to derive diffeomorphism cocycles from Weyl cocycles, with potential implications for effective actions in higher-dimensional conformal field theories.

Abstract

The general structure of trace anomaly, suggested recently by Deser and Shwimmer, is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in $d=2k$ dimensions is considered, and explicit answers for $d=4,6$ are obtained. Particularly, it is shown, that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field $σ$, i.e. the form, similar to the two-dimensional Lioville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order $d$ and zero weight. Beside that, the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group is presented.