  We consider sequences of quadratic non-local functionals, depending on a
small parameter $\e$, that approximate the Dirichlet integral by a well-known
result by Bourgain, Brezis and Mironescu. Similarly to what is done for
hard-core approximations to vortex energies in the case of the Dirichlet
integral, we further scale such energies by $|\log\e|^{-1}$ and restrict them
to $S^1$-valued functions. We introduce a notion of convergence of functions to
integral currents with respect to which such energies are equi-coercive, and
show the converge to a vortex energy, similarly to the limit behaviour of
Ginzburg-Landau energies at the vortex scaling.
