We study the properties of Wilson loops in three dimensional non-compact U(1)
gauge theories with global abelian symmetries. We use duality in the continuum
and on the lattice, to argue that close to the critical point between the Higgs
and Coulomb phases, all correlators of the Wilson loops are periodic functions
of the Wilson loop charge, Q. The period depends on the global symmetry of the
theory, which determines the magnetic flux carried by the dual particles. For
single flavour scalar electrodynamics, the emergent period is Q = 1. In the
general case of N complex scalars with a U(1)^{N-1} global symmetry, the period
is Q = N. We also give some arguments why this phenomenon does not generalize
to theories with a full non-abelian SU(N) symmetry, where no periodicity in Q
is expected. Implications for lattice simulations, as well as for physical
systems, such as easy plane antiferromagnets and disordered superfluids, are
noted.Comment: 25 pages, 1 figur