The noncommutative family Atiyah-Patodi-Singer index theorem

2014/12/09 by Yong Wang, Wang, Yong
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.1412.2870

openalex publication_date 2014/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Noncommutative geometry, conformal geometry, and the local equivariant index theorem
• Volterra calculus, local equivariant family index theorem and equivariant eta forms

Related

• Noncommutative Maslov Index and Eta Forms
• The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras
• Index theorey and Non-Commutative Geometry. II. Dirac operators and index bundles
• Eta forms and the Chern Character
• Bivariant Chern character and the longitudinal index theory