The noncommutative family Atiyah-Patodi-Singer index theorem

TL;DR

This paper extends the noncommutative Atiyah-Patodi-Singer index theorem to families of operators, introducing eta cochain forms, decompositions, and a new family $b$-Chern-Connes character, with proofs of regularity and entireness.

Contribution

It introduces a family version of the noncommutative Atiyah-Patodi-Singer index theorem, including new eta cochain forms and the family $b$-Chern-Connes character.

Findings

Proved regularity of eta cochain form for kernel as a vector bundle.

Decomposed eta form and Chern character as pairings involving idempotents.

Established the entireness and variation formula of the family $b$-Chern-Connes character.

Abstract

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fibration with boundary as a pairing of the family Chern-Connes character for a manifold with boundary with the Chern character of an idempotent matrix. We define the family $b$-Chern-Connes character and then we prove that it is entire and give its variation formula. By this variation formula, we prove another noncommutative family Atiyah-Patodi-Singer index theorem. Thus, we extend the results of Gezler and Wu to the family case.