Conformally Covariant Differential Operators: Properties and Applications

TL;DR

This work develops a general framework for conformally covariant differential operators in arbitrary dimensions, examining how their Green functions and metric variations behave under Weyl rescalings. It provides explicit constructions and flat-space Green functions for second-order operators on k-forms and on Weyl-tensor–like fields, as well as fourth-order scalar operators, all constrained by conformal invariance and inversion geometries. The analysis identifies conformal invariants built from the Weyl tensor, Cotton tensor, and related scalars, and shows how these invariants govern operator variations and Green-function structures. The results point toward higher-dimensional analogues of the Polyakov action, offering a path to non-local effective actions that encode conformal anomalies in curved space and yield conformally invariant correlators in flat space, with notable caveats at critical dimensions such as d=4 and d=6.

Abstract

We discuss conformally covariant differential operators, which under local rescalings of the metric, δ_σg^{μν} = 2 σg^{μν}, transform according to δ_σΔ= r Δσ+ (s-r) σΔfor some r if Δis s-th order. It is shown that the flat space restrictions of their associated Green functions have forms which are strongly constrained by flat space conformal invariance. The same applies to the variation of the Green functions with respect to the metric. The general results are illustrated by finding the flat space Green function and also its first variation for previously found second order conformal differential operators acting on $k$-forms in general dimensions. Furthermore we construct a new second order conformally covariant operator acting on rank four tensors with the symmetries of the Weyl tensor whose Green function is similarly discussed. We also consider fourth order operators, in particular a fourth order operator acting on scalars in arbitrary dimension, which has a Green function with the expected properties. The results obtained here for conformally covariant differential operators are generalisations of standard results for the two dimensional Laplacian on curved space and its associated Green function which is used in the Polyakov effective gravitational action. It is hoped that they may have similar applications in higher dimensions.