We discuss conformally covariant differential operators, which under localrescalings 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 spacerestrictions of their associated Green functions have forms which are stronglyconstrained by flat space conformal invariance. The same applies to thevariation of the Green functions with respect to the metric. The generalresults are illustrated by finding the flat space Green function and also itsfirst variation for previously found second order conformal differentialoperators acting on $k$-forms in general dimensions. Furthermore we construct anew second order conformally covariant operator acting on rank four tensorswith the symmetries of the Weyl tensor whose Green function is similarlydiscussed. We also consider fourth order operators, in particular a fourthorder operator acting on scalars in arbitrary dimension, which has a Greenfunction with the expected properties. The results obtained here for conformally covariant differential operatorsare generalisations of standard results for the two dimensional Laplacian oncurved space and its associated Green function which is used in the Polyakoveffective gravitational action. It is hoped that they may have similarapplications in higher dimensions.
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机译：我们讨论共形协变微分算子，该算子在度量的本地重定标下\ delta_ \ sigma g ^ {\ mu \ nu} = 2 \ sigma g ^ {\ mu \ nu}时根据\ delta_ \ sigma \ Delta = r进行变换如果\ Delta是s阶，则r的\ Delta \ sigma +（sr）\ sigma \ Delta。结果表明，与其相关的格林函数的平坦空间限制具有受平坦空间共形不变性强烈约束的形式。关于度量的Green函数的变化也是如此。通过找到平面空间格林函数及其先前发现的，对一般维数作用于$ k $形式的二阶保形微分算子的第一变型，可以说明一般结果。此外，我们构造了一个新的二阶保形协变算子，其作用于具有绿色函数的Weyl张量的对称性的四阶张量。我们还考虑了四阶算子，尤其是作用于任意维标量的四阶算子，它具有具有预期特性的Green函数。此处为保形协变微分算子获得的结果是二维Laplacian弯曲空间及其在Polyakov有效引力作用中使用的关联Green函数的标准结果的一般化。希望它们在更高的尺寸上有类似的应用。
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