Abelian Higgs vortex moduli space (${\mathcal{M}}_N$)

At critical coupling, the Abelian Higgs vortex moduli space is the space of gauge-equivalence classes of static $N$-vortex solutions. Vortex positions provide complex coordinates, and the field-theory kinetic energy induces an $L^2$ Riemannian metric whose geodesics approximate slow vortex motion.

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.

A head-on geodesic through the smooth coincidence point of the two-vortex moduli space emerges along the orthogonal axis. Two identical critically coupled vortices therefore scatter through $90^{\circ }$ in the slow-motion approximation.