Abstract
We study generalized heat kernel coefficients, which appear in the trace of the heat kernel with an insertion of a first-order differential operator, by using a path integral representation. These coefficients may be used to study gravitational anomalies, i.e. anomalies in the conservation of the stress tensor. We use the path integral method to compute the coefficients related to the gravitational anomalies of theories in a non-abelian gauge background and flat space of dimensions 2, 4, and 6. In 4 dimensions one does not expect to have genuine gravitational anomalies. However, they may be induced at intermediate stages by regularization schemes that fail to preserve the corresponding symmetry. A case of interest has recently appeared in the study of the trace anomalies of Weyl fermions.
Highlights
Heat kernel methods provide a useful tool for investigating QFTs
We report in appendix A the worldline propagators defined by the Dirichlet boundary conditions and by the string inspired method, in appendix B we use them for computing some simple Seeley-DeWitt coefficients as a simple review of the path integral method, and in appendix C we report further calculational details
We have studied path integral methods to compute heat kernel traces with insertion of a firstorder differential operator
Summary
Heat kernel methods provide a useful tool for investigating QFTs. They were introduced by Schwinger for studying QED processes [1] and extended to curved spaces and non-abelian gauge fields by DeWitt [2]. The verification of the same result with a PV Dirac mass could not be completed, as the latter induces gravitational anomalies, which can be computed by using a generalized heat kernel coefficient in the background of a non-abelian gauge field and flat spacetime. This justifies the use of flat space that we consider in our analysis. We report in appendix A the worldline propagators defined by the Dirichlet boundary conditions and by the string inspired method, in appendix B we use them for computing some simple Seeley-DeWitt coefficients as a simple review of the path integral method, and in appendix C we report further calculational details
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