Noncommutative 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 nonunique. A consequence is that Yang-Mills-Higgs theory can be reformulated as a generalized Yang-Mills gauge theory on Euclidean space with a ${\mathbf{Z}}_2$ internal structure. By extending the Hodge star operation to this noncommutative 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.

• Received 29 January 1997

DOI:https://doi.org/10.1103/PhysRevD.56.2291