= Symmetry up to gauge transformation
A spatial transformation is a symmetry of a gauge-field configuration when its pullback lies in the <gauge orbit> of the original configuration. It need not fix a particular representative. For a planar vortex, rotation changes its winding phase and a constant $U(1)$ transformation compensates. For a <hedgehog ansatz for a monopole>, a spatial rotation is compensated by the corresponding internal adjoint $SU(2)$ rotation. Gauge-invariant densities remain invariant without choosing a special gauge.
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