Monopoles, Vortices and Kinks in the Framework of Non-Commutative Geometry
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 Z2 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.
Physical Review D
Teo, E. and TING, Hian Ann, Christopher.
Monopoles, Vortices and Kinks in the Framework of Non-Commutative Geometry. (1997). Physical Review D. 56, 2291-2302. Research Collection Lee Kong Chian School Of Business.
Available at: https://ink.library.smu.edu.sg/lkcsb_research/1876