= Complex potential reduction of anti-self-dual Yang-Mills
{title2=$K_{w\bar w}+K_{z\bar z}-[K_w,K_z]=0$}
= ASDYM potential equation
{c}
{synonym}
Locally, the <ASDYM equations> permit a complex gauge with $A_w=A_z=0$. The trace equation then gives $A_{\bar w}=K_z$ and $A_{\bar z}=-K_w$. The remaining curvature equation is $K_{w\bar w}+K_{z\bar z}-[K_w,K_z]=0$. A complex gauge generally makes $K$ valued in the <complexification of a Lie algebra>; reconstructing a real connection requires its inherited reality condition. The gauge/potential construction is local and need not preserve prescribed behaviour at infinity.
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