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

TL;DR

This paper develops a path integral approach to compute generalized heat kernel coefficients with first-order operator insertions, aiding the study of gravitational anomalies in various dimensions.

Contribution

It introduces a novel path integral method to calculate heat kernel coefficients with operator insertions, relevant for analyzing gravitational anomalies in gauge theories.

Findings

Computed coefficients for 2, 4, and 6 dimensions.

Identified potential induced anomalies in 4D due to regularization.

Provided tools for studying trace anomalies of Weyl fermions.

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.