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 $$\partial_{tt} u - \Delta u = \Omega \cdot du + F(u) \quad \text{in $\mathbb{R}^d \times \mathbb{R}$}$$ where $d \geq 4$ and $\Omega$ is a suitable antisymmetric potential. We show that the assumptions on $\Omega$ 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 $\dot{H}^{\frac{d}{2}}(\mathbb{R}^d)$.