Connes-Chern character for manifolds with boundary and eta cochains

TL;DR

This paper develops a new expression for the Connes-Chern character on manifolds with boundary, linking it to eta cochains and relative cyclic cohomology, and explores geometric implications and boundary-related K-theory pairings.

Contribution

It introduces a novel cocycle expression for the Connes-Chern character involving eta cochains, extending the understanding of boundary effects in noncommutative geometry.

Findings

Reveals a new cocycle expression depending on a scaling parameter.

Shows the pairing captures boundary information in K-theory.

Demonstrates the restriction of the Atiyah-Patodi-Singer pairing to almost flat bundles.

Abstract

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.