  We investigate a fractional notion of gradient and divergence operator. We
generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional
div-curl quantities, obtaining, in particular, a nonlocal version of Wente's
lemma.
  We demonstrate how these quantities appear naturally in nonlocal geometric
equations, which can be used to obtain a theory for fractional harmonic maps
analogous to the local theory. Firstly, regarding fractional harmonic maps into
spheres, we obtain a conservation law analogous to Shatah's conservation law
and give a new regularity proof analogous to H\'elein's for harmonic maps into
spheres.
  Secondly, we prove regularity for solutions to critical systems with nonlocal
antisymmetric potentials on the right-hand side. Since the half-harmonic map
equation into general target manifolds has this form, as a corollary, we obtain
a new proof of the regularity of half-harmonic maps into general target
manifolds following closely Rivi\`{e}re's celebrated argument in the local
case.
  Lastly, the fractional div-curl quantities provide also a new, simpler, proof
for H\"older continuity of $W^{s,n/s}$-harmonic maps into spheres and we extend
this to an argument for $W^{s,n/s}$-harmonic maps into homogeneous targets.
This is an analogue of Strzelecki's and Toro-Wang's proof for $n$-harmonic maps
into spheres and homogeneous target manifolds, respectively.
