Geometric Classification of Conformal Anomalies in Arbitrary Dimensions

TL;DR

This paper provides a comprehensive geometric framework for understanding conformal anomalies across all even dimensions, classifying them into two distinct types based on their origin and properties.

Contribution

It introduces a complete geometric classification of conformal anomalies in arbitrary even dimensions, distinguishing between scale-free and scale-dependent classes and providing explicit examples.

Findings

Two classes of conformal anomalies identified: scale-free and scale-dependent.

Anomalies related to topological Euler density and local Weyl invariants.

Explicit examples in dimensions 2, 4, and 6 provided.

Abstract

We give a complete geometric description of conformal anomalies in arbitrary, (necessarily even) dimension. They fall into two distinct classes: the first, based on Weyl invariants that vanish at integer dimensions, arises from finite -- and hence scale-free -- contributions to the effective gravitational action through a mechanism analogous to that of the (gauge field) chiral anomaly. Like the latter, it is unique and proportional to a topological term, the Euler density of the dimension, thereby preserving scale invariance. The contributions of the second class, requiring introduction of a scale through regularization, are correlated to all local conformal scalar polynomials involving powers of the Weyl tensor and its derivatives; their number increases rapidly with dimension. Explicit illustrations in dimensions 2, 4 and 6 are provided.