Appearance of vortices and monopoles in a decomposition of an SU(2) Yang–Mills field

A decomposition for the SU(2) Yang–Mills field in the low-energy limit is obtained by supposing that, in the low-energy limit, the field strength tensor of an SU(2) Yang–Mills field can be obtained by multiplying two parts, and , such that . is a space-time tensor and is an isotriplet unit vector field that gives the Abelian direction at each space-time point. By Abelianizing the field strength tensor , we show how (singular) vortices and monopoles can appear in the low-energy limit of SU(2) Yang–Mills theory. If the decomposition is valid on the boundary of the system, then we show that vortices with finite string tension can also appear. The interesting point is that we have started with a decomposed Yang–Mills field and we have ended up with a theory with an Abelian gauge field coupled to a scalar field. The effect of this scalar field on monopoles is also discussed