  We consider soliton solutions to Bekenstein's theory, for which the fine
structure constant $\alpha=e^2/(4\pi\hbar c)$ is allowed to vary due to the
presence of a dielectric field pervading the vacuum. More specifically we
investigate the effects of a varying $\alpha$ upon a complex scalar field with
a U(1) electromagnetic gauge symmetry subject to spontaneous symmetry breaking.
We find vortex solutions to this theory, similar to the Nielsen-Olesen vortex.
Near the vortex core the electric charge is typically much larger than far away
from the string, lending these strings a superconducting flavour. In general
the dielectric field coats the usual local string with a global string
envelope. We discuss the cosmological implications of networks of such strings,
with particular emphasis on their ability to generate inhomogeneous
recombination scenarios. We also consider the possibility of the dielectric
being a charged free field. Even though the vacuum of such a field is trivial,
we find that the dielectric arranges itself in the shape of a local string,
with a quantized magnetic flux at the core -- presumably borrowing these
topological features from the underlying Nielsen-Olesen vortex.
