Weyl vs. Conformal

TL;DR

This work clarifies the distinction between conformal and Weyl invariance and investigates when conformally invariant flat-space theories can be promoted to Weyl-invariant theories upon coupling to gravity. By analyzing higher-derivative scalar theories in both flat and curved spacetimes, it uses the Paneitz operator, GJMS classifications, and curvature tensors (Schouten, Bach) to test Weyl covariance. The main findings show that Weyl invariance is strictly stronger than conformal invariance: many flat-space conformal theories cannot be Weyl-invariant in four dimensions (e.g., $\Box^3$) due to obstructions, while some higher-derivative conformal operators have well-defined Weyl-covariant generalizations only under specific dimensional and geometric conditions. The results provide criteria for when conformally invariant theories admit Weyl-invariant gravitational couplings, highlighting the role of Einstein manifolds and global versus full conformal invariance in reduced dimensions.

Abstract

In this note we show that given a conformally invariant theory in flat space-time, it is not always possible to couple it to gravity in a Weyl invariant way.