Eta forms and the Chern Character

TL;DR

This paper establishes two geometric index theorems for elliptic operator families on manifolds with boundary, utilizing eta forms as representatives of Chern character classes in the index bundle.

Contribution

It introduces a novel approach to computing eta form representatives for Chern characters in the context of boundary value problems for elliptic operators.

Findings

Proved two geometric index theorems involving eta forms

Computed eta form representatives for Chern characters

Connected eta forms with superconnection character forms

Abstract

We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limiting values of superconnection character forms.