Vortices as Degenerate Metrics

We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian–Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate Hermitian metrics that satisfy a certain curvature equation. Using this viewpoint, we rephrase standard results about vortices and make new observations. We note the existence of a conceptually simple, non-linear rule for superposing vortex solutions, and we describe the natural behaviour of the L ²-metric on the moduli space upon restriction to a class of submanifolds.