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

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.

28 pages, Latex

References in corpus (8)

• The Spectral Action Principle
• Self-Dual Chern-Simons Theories
• A Universal Action Formula
• The Standard Model as a noncommutative geometry: the low energy regime
• Gravity and Electromagnetism in Noncommutative Geometry
• Higgs Mass and Noncommutative Geometry
• BRST Quantization of Gauge Theory in Noncommutative Geometry: Matrix Derivative Approach
• $N=2$ and $4$ Super Yang-Mills Theories on $M_4\times Z_2\times Z_2$ Geometry

Cited by in corpus (5)

• Trieste lectures on solitons in noncommutative gauge theories
• Noncommutative instantons revisited
• Magnetic monopole bibliography
• Non Abelian Vortices as Instantons on Noncommutative Discrete Space
• Equivariance on Discrete Space and Yang-Mills-Higgs Model