Conformally covariant differential operators: Properties and applications

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 of order s. It is shown that the fat 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 generalizations 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.