Monopoles, vortices and kinks in the framework of non-commutative geometry

TL;DR

This paper explores how non-commutative geometry reformulates gauge theories, revealing that monopoles, vortices, and kinks are self-dual solutions in specific dimensions, and establishes the dimensional constraints for static solitons.

Contribution

It introduces a unified non-commutative geometric framework for gauge theories, connecting soliton solutions to self-duality and dimensional restrictions.

Findings

Monopoles, vortices, and kinks are self-dual solutions in their respective dimensions.

Static solitons exist only in one, two, and three spatial dimensions.

Reformulation of Yang-Mills-Higgs theory as a generalized gauge theory on non-commutative space.

Abstract

Non-commutative differential geometry allows a scalar field to be regarded as a gauge connection, albeit on a discrete space. We explain how the underlying gauge principle corresponds to the independence of physics on the choice of vacuum state, should it be non-unique. A consequence is that Yang-Mills-Higgs theory can be reformulated as a generalised Yang-Mills gauge theory on Euclidean space with a $Z_2$ internal structure. By extending the Hodge star operation to this non-commutative space, we are able to define the notion of self-duality of the gauge curvature form in arbitrary dimensions. It turns out that BPS monopoles, critically coupled vortices, and kinks are all self-dual solutions in their respective dimensions. We then prove, within this unified formalism, that static soliton solutions to the Yang-Mills-Higgs system exist only in one, two and three spatial dimensions.