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We study generalized heat kernel coefficients, which appear in the trace ofthe heat kernel with an insertion of a first-order differential operator, byusing a path integral representation. These coefficients may be used to studygravitational anomalies, i.e. anomalies in the conservation of the stresstensor. We use the path integral method to compute the coefficients related tothe gravitational anomalies of theories in a non-abelian gauge background andflat space of dimensions 2, 4, and 6. In 4 dimensions one does not expect tohave genuine gravitational anomalies. However, they may be induced atintermediate stages by regularization schemes that fail to preserve thecorresponding symmetry. A case of interest has recently appeared in the studyof the trace anomalies of Weyl fermions.

Path integral calculation of heat kernel traces with first order operator insertions