The noncommutative family Atiyah-Patodi-Singer index theorem

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.

25 pages, to appear in Journal of Geometry and Physics

References in corpus (4)

• Noncommutative geometry, conformal geometry, and the local equivariant index theorem
• Volterra calculus, local equivariant family index theorem and equivariant eta forms
• The noncommutative infinitesimal equivariant index formula: part II
• The Noncommutative Infinitesimal Equivariant Index Formula