Neumann Boundary Condition for Abelian Vortices

We study abelian BPS vortices on a surface $S$ with boundary, which satisfy the Neumann boundary condition on the norm of the scalar field, or equivalently, that the current along the boundary vanishes. These vortices have quantised magnetic flux and quantised energy. Existence of such vortices is manifest when $S$ is the quotient by a reflection of a smooth surface without boundary, for example a hemisphere. The $N$-vortex moduli space then admits an interesting stratification, depending on the number of vortices in the interior of $S$ and the number of half-vortices on the boundary.