Vector Higgs field ($\Phi \in {\mathbb{R}}^3$)

In the SO(3) vector Higgs model, a real scalar field $\Phi \in {\mathbb{R}}^3$ transforms as $\Phi \mapsto R\Phi$. Its gauge covariant derivative is $D_{\mu }\Phi ={\partial }_{\mu }\Phi +g{\mathbf{A}}_{\mu }\times \Phi$ for generators acting by the cross product.

If $D_{\mu }\Phi =0$ in the SO(3) vector Higgs model, antisymmetry of the gauge potential implies ${\partial }_{\mu }|\Phi {|}^2=0$. On a local patch with nonzero constant norm, unitary gauge aligns the field with a fixed axis. The connection and gauge field strength then take values in the axis stabilizer; curvature can remain nonzero. Global alignment can require several bundle patches.