Weyl-Gauging and Conformal Invariance
A. Iorio, L. O'Raifeartaigh, I. Sachs, C. Wiesendanger
TL;DR
The paper establishes a precise link between scale, Weyl, and conformal invariance by promoting rigid Weyl-invariance to a local gauge and then showing that Weyl gauging can be replaced by curvature couplings (Ricci gauging) exactly when the flat-space theory is conformally invariant. It provides an algebraic criterion based on the virial current, derives the improved energy-momentum tensor systematically, and classifies scale-invariant Lagrangians that are conformally invariant for fields of arbitrary spin. In two dimensions, it connects conformal invariance, gravitational anomalies, and the Virasoro central charge, while in higher dimensions it shows that conformally invariant theories admit Ricci gauging with couplings to the Ricci tensor and scalar. A concrete counterexample demonstrates that scale invariance does not always imply conformal invariance, establishing the precise conditions under which these symmetries align and how they shape the energy-momentum structure across spins.
Abstract
Scale-invariant actions in arbitrary dimensions are investigated in curved space to clarify the relation between scale-, Weyl- and conformal invariance on the classical level. The global Weyl-group is gauged. Then the class of actions is determined for which Weyl-gauging may be replaced by a suitable coupling to the curvature (Ricci gauging). It is shown that this class is exactly the class of actions which are conformally invariant in flat space. The procedure yields a simple algebraic criterion for conformal invariance and produces the improved energy-momentum tensor in conformally invariant theories in a systematic way. It also provides a simple and fundamental connection between Weyl-anomalies and central extensions in two dimensions. In particular, the subset of scale-invariant Lagrangians for fields of arbitrary spin, in any dimension, which are conformally invariant is given. An example of a quadratic action for which scale-invariance does not imply conformal invariance is constructed.