Let $(M,g)$ be a riemannian manifold and let $P\to M$ be a principal $G$-bundle with connection $A$. Let $\alpha \in \Omega^1(M;\mathrm{ad}P)$ be a one-form on $M$ with values in the adjoint bundle $\mathrm{ad} P$. Then consider the equation $$ F_{A + \alpha} = \tfrac12 [\alpha,\alpha] $$ in $\Omega^2(M;\mathrm{ad}P)$, where $F_{A+ \alpha}$ is the curvature of the connection $A + \alpha$.

This equation is reminiscent of the Hitchin equations on a Riemann surface, but of course is different.

(In the context in which I have seen this equation, it is supplemented by the condition $\nabla^A \alpha \in \Omega^2(M;\mathrm{ad}P)$, which is equivalent to the vanishing of the symmetrisation of $\nabla^A \alpha$.)

**Question**

Has anyone come across such an equation before? And if so, could you point me to a reference?

dontmean just the second order obstruction to deforming a flat connection? – Paul Jul 12 '11 at 20:26