The Yang-Mills equation for a connection 
on a principal bundle
over a Riemannian manifold is
 |
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
 |
The Yang-Mills existence and mass gap problem, one of the Millennium Prize Problems, asks for a rigorous construction of the corresponding quantum field theory with a positive mass gap. Solving the classical Yang-Mills equation alone does not settle this quantum existence problem (Jaffe and Witten).
