The relaxed energy of fractional Sobolev maps with values into the circle

We deal with the weak sequential density of smooth maps in the fractional Sobolev classes of W-s,W-p maps in high dimension domains and with values into the circle. When s is lower than one, using interpolation theory we introduce a natural energy in terms of optimal extensions on suitable weighted Sobolev spaces. The relaxation problem is then discussed in terms of Cartesian currents. When sp=1, the energy gap of the relaxed functional is always finite and is given by the minimal connection of the singularities times an energy weight, obtained through a minimum problem for one dimensional W-1/p,W-p maps with degree one. When sp>1, instead, concentration on codimension one sets needs unbounded energy. We finally treat the case where s is greater than one, obtaining an almost complete picture.