Topological singularities arising from fractional-gradient energies

We prove that, on a planar regular domain, suitably scaled functionals of Ginzburg–Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, Γ-converge to vortex-type energies with respect to the flat convergence of Jacobians. The compactness and the Γ-liminf follow by comparison with standard Ginzburg–Landau functionals depending on Riesz potentials. The Γ-lim sup, instead, is achieved via a direct argument by joining a finite number of vortex-like functions suitably truncated around the singularity.