Relative coordinate for two identical vortices ($w=(z_1-z_2{)}^2$)

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.