= Axial transformation of orthogonal gauge connections
{title2=$R[A]_\times R^{-1}=[(\det R)RA]_\times$}
For an <orthogonal group> gauge transformation $R\in O(3)$, a scalar vector transforms as $\phi\mapsto R\phi$. The <gauge connection> belongs to the <Adjoint representation>, identified with axial vectors: the displayed cross-product identity makes its homogeneous transformation $A\mapsto(\det R)RA$. The matrix connection also has the usual inhomogeneous derivative term. Thus $D\phi$ transforms as an ordinary vector, including under improper orthogonal transformations. Treating both fields as ordinary vectors would fail because a <cross product> changes sign under an orientation reversal. A nonzero vector vacuum has <stabilizer subgroup> $O(2)$; its connected part is $SO(2)$.
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