GEOMETRIC CLASSIFICATION OF CONFORMAL ANOMALIES IN ARBITRARY DIMENSIONS

We give a complete geometric description of conformal anomalies in arb itrary, (necessarily even) dimension. They fall into two distinct clas ses: the first, based on Weyl invariants that vanish at integer dimens ions, 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 dimensio n, thereby preserving scale invariance. The contributions of the secon d 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 rapidl y with dimension. Explicit illustrations in dimensions 2, 4 and 6 are provided.