Closed Form Effective Conformal Anomaly Actions in D$\geq$4

TL;DR

The paper tackles constructing closed-form effective actions for conformal anomalies in dimensions four and higher, focusing on type B anomalies with logarithmic cutoff dependence. It introduces a novel class of Weyl-invariant tensor operators and uses them to derive a compact closed-form type B action in four dimensions, such as an action that yields the correct C^2 variation under Weyl transformations. It shows that naive higher-dimensional Type A generalizations based on the Paneitz operator lead to double-pole nonlocalities and ghost-like compensator dynamics, rendering them physically unsatisfactory beyond leading order. The work discusses ambiguities and potential improvements, notably the existence of a suitable higher-rank operator (tilde-Delta) that would allow covariant, physically acceptable Type A constructions. The results provide a clearer separation between Type B anomalies, which can be captured by robust nonlocal Weyl-invariant actions, and Type A anomalies, which require further development, with implications for holography and related aspects of quantum gravity.

Abstract

I present, in any D$\geq$4, closed-form type B conformal anomaly effective actions incorporating the logarithmic scaling cutoff dependence that generates these anomalies. Their construction is based on a novel class of Weyl-invariant tensor operators. The only known type A actions in D$\geq$4 are extensions of the Polyakov integral in D=2; despite contrary appearances, we show that their nonlocality does not conflict with general anomaly requirements. They are, however, physically unsatisfactory, prompting a brief attempt at better versions.