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 YangMills equations as: $$ R(\nabla, \nabla')=0$$ $$R(\nabla)=R(\nabla').$$ For $\nabla=\nabla'$, we have the usual YangMills equations. Can we have moduli spaces over a riemannian surface? Have we symplectic reduction?
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7$\begingroup$ If $\nabla = \nabla'$, what you call the YM equations is simply the flatness condition. This is not what I normally call YangMills. Could you perhaps elaborate? $\endgroup$ – José FigueroaO'Farrill Aug 4 '16 at 18:03

$\begingroup$ Yes, I call the YangMills 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 LeviCivita connections. The interest seems so to be rather vast. $\endgroup$ – Antoine Balan Aug 4 '16 at 20:19