Instanton algebras and quantum 4-spheres

TL;DR

This paper explores generalized instanton algebras on quantum 4-spheres, describing complex rank 2 bundles and analyzing their topological invariants in a noncommutative geometric setting.

Contribution

It introduces a framework for instantons on quantum 4-spheres and clarifies the role of Chern-Connes classes in this noncommutative context.

Findings

Quantum 4-spheres $S^4_q$ are derived from natural instanton conditions.

Instantons are represented by self-adjoint idempotents e.

The paper clarifies the vanishing of $ch_1(e)$ and the use of $ch_2(e)$ as a volume form.

Abstract

We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\in\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.