A decomposition for SU(2) Yang-Mills fields

Motivated by Abelian dominance, we suppose that the field strength tensor in the low energy limit of the SU(2) Yang-Mills theory is $ G_{\mu\nu}=G_{\mu\nu} n $, where $ G_{\mu\nu} $ is a space-time tensor and $ n $ is a unit vector field which selects the Abelian direction at each space-time point. Based on this form of the field strength tensor, we propose a decomposition for the Yang-Mills field with three degrees of freedom. It seems that by this kind of decompostion, both monopoles and vortices appear at the same time. We have also obtained the Dirac quantization condition with a rescaled electric charge.