The Yang-Mills equation for a connection 
on a principal bundle
over a Riemannian manifold is
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where 
is the bundle curvature, 
is the exterior derivative coupled to 
, and 
is its formal adjoint. It is the Euler-Lagrange
differential equation obtained by varying the action functional
of Yang-Mills theory. In four dimensions, a
self-dual or anti-self-dual bundle
curvature automatically satisfies the Yang-Mills
equation. In suitable local complex coordinates, the anti-self-dual Yang-Mills
equation can be written as the system of partial
differential equations
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