= Relative coordinate for two identical vortices
{title2=$w=(z_1-z_2)^2$}
Interchanging two <Abelian Higgs vortices> replaces $q=z_1-z_2$ by $-q$. Thus $w=q^2$ is a single-valued relative coordinate on the unordered <Abelian Higgs vortex moduli space>. Coincidence is smooth in $w$; a geodesic crossing from positive real $w$ to negative real $w$ changes the line of separation by $\pi/2$, explaining <right-angle scattering of Abelian Higgs vortices>. The asymptotic cone in the $q$ coordinate has angular period $\pi$ and must not be treated as a singularity of the actual coincidence metric.
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