Dynamical Noncommutative Spheres

TL;DR

This paper introduces a new family of noncommutative 4-spheres with trivial first Chern class and explicitly computes their Chern-Connes pairing, advancing the understanding of noncommutative geometry and instanton bundles.

Contribution

It constructs a family of noncommutative 4-spheres with specific topological properties and explicitly calculates their Chern-Connes pairing, providing new examples in noncommutative geometry.

Findings

Defined a family of noncommutative 4-spheres with trivial first Chern class

Constructed a 4-dimensional cycle for these spheres

Explicitly calculated the Chern-Connes pairing for the instanton projector

Abstract

We introduce a family of noncommutative 4-spheres, such that the instanton projector has its first Chern class trivial: $ch_1(e) = B \chi + b \xi$. We construct for them a 4-dimensional cycle and calculate explicitly the Chern-Connes paring for the instanton projector.