Consider a vector bundle, $E \to M$, with connection, $\nabla$, and curvature $2$-form, $F$ on $M$. For $E$-valued differential forms on $M$, $\Omega(M, E)$, we have an exterior covariant derivative, $d_{\nabla}$ for which $d_{\nabla}^2 = F \wedge (-)$.

Question: What is the significance of the collection of forms, $$\{ \omega \in \Omega(M, E) \ | \ F \wedge \omega = 0 \}?$$