The noncommutative infinitesimal equivariant index formula: part II

TL;DR

This paper establishes the well-definedness of infinitesimal equivariant Chern-Connes characters and eta cochains, computes their limits, and extends the noncommutative index formula to manifolds with boundary, advancing noncommutative geometry.

Contribution

It introduces a new framework for infinitesimal equivariant index theory, including boundary cases, using Getzler symbol calculus and pairing techniques.

Findings

Infinitesimal equivariant Chern-Connes characters are well-defined.

The limit of these characters as time approaches zero is computed.

The noncommutative index formula is extended to manifolds with boundary.

Abstract

In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes characters when the time goes to zero by using the Getzler symbol calculus and then extend these theorems to the family case. We also prove that infinitesimal equivariant eta cochains are well-defined and prove the noncommutative infinitesimal equivariant index formula for manifolds with boundary.