Some Modifications of Getzler's Grading technique

TL;DR

This paper reviews and adapts Getzler's grading technique to compute leading terms in heat kernel asymptotic expansions for different mathematical contexts.

Contribution

It introduces modifications to Getzler's grading method for broader applications in heat kernel analysis.

Findings

Adapted grading technique for new heat kernel scenarios

Computed leading asymptotic terms in different settings

Enhanced understanding of local index theorem applications

Abstract

This paper reviews the grading technique developed by Getzler to prove the local index theorem and shows how to adapt it to compute the leading terms of asymptotic expansions of traces of heat kernels in two other situations.