On wave systems with antisymmetric potential in dimension d >= 4 and well-posedness for (half-)wave maps

We prove a priori estimates for wave systems of the type $${\mathrm{\partial }}_{tt}u-\mathrm{\Delta }u=\mathrm{\Omega }\cdot du+F(u)\text{in }{\mathbb{R}}^d\times \mathbb{R}$$ where $d\ge 4$ and $\mathrm{\Omega }$ is a suitable antisymmetric potential. We show that the assumptions on $\mathrm{\Omega }$ are applicable to wave- and half-wave maps, the latter by means of the Krieger-Sire reduction. We thus obtain well-posedness of those equations for small initial data in ${\dot{H}}^{\frac{d}{2}}({\mathbb{R}}^d)$.