Heat Kernel Asymptotics on Homogeneous Bundles

TL;DR

This paper develops a formal solution for the heat kernel diagonal on homogeneous bundles over symmetric spaces, providing a generating function for heat invariants and demonstrating its correctness through regularization and continuation.

Contribution

It introduces a novel integral representation approach to derive heat kernel asymptotics on homogeneous bundles over symmetric spaces.

Findings

Formal solution reproduces heat kernel diagonal after regularization

Generates entire sequence of heat invariants

Validates approach through analytical continuation

Abstract

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.