Singular vortex pairs follow magnetic geodesics

We consider pairs of point vortices having circulations ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $\epsilon$, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the ``singular vortex pair" moves as a single charged particle on the surface with a charge of order $1\mathrm{/}{\epsilon }^2$ in a magnetic field $B$ which is everywhere normal to the surface and of strength $\mathrm{\mid }B\mathrm{\mid }={\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2$. In the case ${\mathrm{\Gamma }}_1=-{\mathrm{\Gamma }}_2$, this gives another proof of Kimura’s conjecture (Kimura 1999) that singular dipoles follow geodesics.