= Skyrmion stabilizer and collective-coordinate orbit
{c}
{title2=$\mathcal O=G/H$}
= Skyrmion collective-coordinate orbit
{c}
{synonym}
For a centered static <Skyrmion> $U_0$, the effective rotation-isorotation group is $G=SO(3)_{\rm space}\times SO(3)_{\rm iso}$. Its <stabilizer subgroup> $H$ consists of pairs obeying $A U_0(R^{-1}x)A^\dagger=U_0(x)$. Thus distinct rigid orientations form the <homogeneous space> $G/H$, by the <orbit-stabilizer theorem>. The unit hedgehog has diagonal $SO(3)$ stabilizer and three orientation coordinates; the toroidal two-Skyrmion has a one-dimensional continuous stabilizer and five; the tetrahedral and cubic cases have discrete stabilizers and six. Adding three translations gives zero-mode orbit dimensions 6, 8, 9, 9. These are symmetry-generated low-energy coordinates, not a claim that non-Bogomolny multi-Skyrmion solutions possess an exact flat moduli space of arbitrary separations.
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