  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
$\dot{H}^{\frac{d}{2}}(\mathbb{R}^d)$.
