Singular vortex pairs follow magnetic geodesics

We consider pairs of point vortices having circulations $Γ_1$ and $Γ_2$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $\varepsilon$, 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 $1/\varepsilon^2$ in a magnetic field $B$ which is everywhere normal to the surface and of strength $|B|=Γ_1 +Γ_2$. In the case $Γ_1=-Γ_2$, this gives another proof of Kimura's conjecture (Kimura 1999) that singular dipoles follow geodesics.