Anti-self-dual Yang-Mills equations (ASDYM equations)

The anti-self-dual Yang-Mills equations require the curvature of a connection on an oriented Euclidean four-manifold to satisfy $F=-\ast F$. In complex coordinates $(w,z)$ they can be written

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ and $A_{\tau }=0$, one anti-self-duality convention becomes