Trace Anomalies and Cocycles of Weyl and Diffeomorphisms Groups

TL;DR

This paper explores the structure of trace anomalies and cocycles related to Weyl and diffeomorphism groups, providing explicit results in four and six dimensions and linking cocycles to conformal invariants.

Contribution

It demonstrates how trace anomalies arise from the Wess-Zumino consistency condition and derives explicit cocycle forms in higher dimensions, connecting them to conformal operators.

Findings

Explicit cocycle formulas for d=4,6 dimensions.

Addition of counterterms simplifies cocycle form.

Connection established between cocycles and conformal-invariant operators.

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 $\sigma$, 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.