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.

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.