The Noncommutative Topology of Anti-Self-Dual Gauge Fields

TL;DR

This paper explores the topology of instanton gauge fields on noncommutative four-spheres using $C^*$-algebra techniques, deforming classical constructions to quantum settings and revealing how classical parameters can be recovered through internal gauge choices.

Contribution

It introduces a method to construct families of instantons on noncommutative spheres via deformation of the ADHM construction, linking noncommutative topology with gauge theory.

Findings

Families of instantons parametrized by noncommutative spaces

Classical parameter spaces can be recovered through internal gauge transformations

Extension of classical instanton constructions to noncommutative geometries

Abstract

Through techniques afforded by $C^*$-algebras and Hilbert modules, we study the topology of spaces which parametrize families of instanton gauge fields on noncommutative Euclidean four-spheres $S^4_\sigma$. By deforming the ADHM construction of instantons on the classical sphere $S^4$, we obtain families of instantons on the quantum sphere which are naturally parametrized by noncommutative topological spaces. Using the internal gauge theory of $S^4_\sigma$ determined by the inner automorphisms of its function algebra, we find that one may always recover a classical parameter space by making a suitable choice of internal gauge.