Comparison of two equivariant $\eta$-forms

TL;DR

This paper introduces a new equivariant infinitesimal ta-form, compares it with the existing form, and analyzes its singular behavior, extending previous results and aiding in localization of ta-invariants and differential K-theory.

Contribution

It defines the equivariant infinitesimal ta-form and compares it with the equivariant ta-form, providing a locally computable form and analyzing singular behavior.

Findings

Comparison of two equivariant ta-forms modulo exact forms

Description of the singular behavior of the ta-form as a function on the Lie group

Extension of Goette's result to broader contexts

Abstract

In this paper, we first define the equivariant infinitesimal $\eta$-form, then we compare it with the equivariant $\eta$-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the equivariant $\eta$-form, modulo exact forms, as a function on the acting Lie group. This result extends a result of Goette and it plays an important role in our recent work on the localization of $\eta$-invariants and on the differential $K$-theory.