A fractional version of Rivi\`ere's GL(N)-gauge

We prove that for antisymmetric vectorfield $\Omega$ with small $L^2$-norm there exists a gauge $A \in L^\infty \cap \dot{W}^{1/2,2}(\mathbb{R}^1,GL(N))$ such that ${\rm div}_{\frac12} (A\Omega - d_{\frac{1}{2}} A) = 0$. This extends a celebrated theorem by Rivi\`ere to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.