  The non-compact lattice version of the Abelian Higgs model is studied in
terms of its topological excitations. The Villain form of the partition
function is represented as a sum over world-sheets of gauge-invariant
``vortex'' strings. The phase transition of the system is then related to the
density of these excitations. Through Monte Carlo simulations the density of
the vortex sheets is shown to be a good order parameter for the system. The
vortex density essentially vanishes in the Higgs phase, and the Coulomb phase
consists of a single vortex condensate.
