Singular Vortex Pairs Follow Magnetic Geodesics

We consider pairs of point vortices having circulations T1 and T2 and confined to a two-dimensional surface S. In the limit of zero initial separation ε, 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/ε² in an magnetic field B that is everywhere normal to the surface and of strength |B| = T1 + T2. In the case T1 = −T2, this gives another proof of Kimura’s conjecture [11] that singular dipoles follow geodesics.