Conformally Covariant Differential Operators: Properties and Applications

TL;DR

This paper explores properties of conformally covariant differential operators, analyzing their Green functions and variations, and introduces new operators with potential applications in higher-dimensional conformal geometry and physics.

Contribution

It provides a comprehensive analysis of conformally covariant differential operators, including new operators on tensors and scalars, with explicit Green functions and variations in arbitrary dimensions.

Findings

Green functions are strongly constrained by conformal invariance.

Explicit Green functions for second and fourth order operators are derived.

New conformally covariant operators on tensors are constructed.

Abstract

We discuss conformally covariant differential operators, which under local rescalings of the metric, \delta_\sigma g^{\mu\nu} = 2 \sigma g^{\mu\nu}, transform according to \delta_\sigma \Delta = r \Delta \sigma + (s-r) \sigma \Delta for some r if \Delta is 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.