Let be over a manifold $M$ two connections $\nabla , \nabla'$. We define the curvature of the two connections as the following tensor: $$R(\nabla,\nabla ') (X,Y)= \nabla_X \nabla'_Y - \nabla_Y \nabla'_X + \nabla'_X \nabla_Y - \nabla'_Y \nabla_X - \nabla_{[X,Y]} - \nabla'_{[X,Y]}$$ We have $$R(\nabla, \nabla')=4 R(\frac{1}{2}(\nabla+ \nabla'))-R(\nabla)-R(\nabla')$$$$R (\nabla, \nabla)= 2 R(\nabla),$$ $R(\nabla)$ being the usual curvature. We can now define the twin Yang-Mills equations as: $$ R(\nabla, \nabla')=0$$ $$R(\nabla)=R(\nabla').$$ For $\nabla=\nabla'$, we have the usual Yang-Mills equations. Can we have moduli spaces over a riemannian surface? Have we symplectic reduction?

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    $\begingroup$ If $\nabla = \nabla'$, what you call the YM equations is simply the flatness condition. This is not what I normally call Yang-Mills. Could you perhaps elaborate? $\endgroup$ – José Figueroa-O'Farrill Aug 4 '16 at 18:03
  • $\begingroup$ Yes, I call the Yang-Mills equations the flatness condition. Following the theorem of Narashiman and Seshadri proved by Donaldson, we obtain moduli spaces of fiber bundles over a Riemann surface. $\endgroup$ – Antoine Balan Aug 4 '16 at 19:27
  • $\begingroup$ We can find other applications of the curvature of two connections, we can take two metrics over a Riemannian manifold and so two Levi-Civita connections. The interest seems so to be rather vast. $\endgroup$ – Antoine Balan Aug 4 '16 at 20:19

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