  In this paper we extend, for the first time, part of the Weierstrass extremal
field theory in the Calculus of Variations to a nonlocal framework. Our model
case is the energy functional for the fractional Laplacian (the
Gagliardo-Sobolev seminorm), for which such a theory was still unknown.
  We build a null-Lagrangian and a calibration for nonlinear equations
involving the fractional Laplacian in the presence of a field of extremals.
Thus, our construction assumes the existence of a family of solutions to the
Euler-Lagrange equation whose graphs produce a foliation. Then, the minimality
of each leaf in the foliation follows from the existence of the calibration. As
an application, we show that monotone solutions to fractional semilinear
equations are minimizers.
  In a forthcoming work we generalize the theory to a wide class of nonlocal
elliptic functionals and give an application to the viscosity theory.
