Higher spectral flow and an entire bivariant JLO cocycle

TL;DR

This paper introduces a new bivariant JLO cocycle associated with smooth fibrations and Dirac operators, proving its entireness under various topologies and linking it to higher spectral flow Chern characters.

Contribution

It defines a bivariant JLO cocycle for families of Dirac operators and proves its entireness in multiple topologies, connecting it to higher spectral flow.

Findings

The bivariant JLO cocycle is entire for $C^{ ext{ell+1}}$ and Fréchet topologies.

It computes the Chern character of the Dai-Zhang higher spectral flow.

The cocycle's entireness extends to various smooth topologies.

Abstract

Given a smooth fibration of closed manifolds and a family of generalised Dirac operators along the fibers, we define an associated bivariant JLO cocycle. We then prove that, for any $\ell \geq 0$, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the $C^{\ell+1}$ topology and functions on the base manifold with the $C^\ell$ topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fr\'echet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.