The Noncommutative Infinitesimal Equivariant Index Formula

TL;DR

This paper develops an infinitesimal equivariant index formula within noncommutative geometry, utilizing heat kernel asymptotics, and introduces related eta cochains and forms for odd-dimensional manifolds.

Contribution

It introduces a novel infinitesimal equivariant index formula in noncommutative geometry, extending to odd dimensions and defining new eta cochains and forms.

Findings

Established an infinitesimal equivariant index formula using heat kernel asymptotics.

Defined and proved regularity of infinitesimal equivariant eta cochains.

Compared infinitesimal and equivariant eta forms, establishing their relationship.

Abstract

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.