  It is shown how to treat the degrees of freedom of Nielsen-Olesen vortices in
the $3+1$-dimensional $U(1)$ higgs model by a collective coordinate method. In
the london limit, where the higgs mass becomes infinite, the gauge and
goldstone degrees of freedom are integrated out, resulting in the vortex
world-sheet action. Introducing an ultraviolet cut-off mimics the effect of
finite higgs mass. This action is non-polynomial in derivatives and depends on
the extrinsic curvature of the surface. Flat surfaces are stable if the
coherence length is less than the penetration depth. It is argued that in the
quantum abelian higgs model, vortex world-sheets are dominated by branched
polymers.
