I define equations like the Seiberg-Witten equations for forms of a riemannian manifold $(M,g)$. $\alpha \in \Lambda_+(TM)$ and $\theta \in \Lambda^1(TM)$.
$$
d\alpha+\theta \wedge \alpha=0
$$
$$
d\theta_+=\frac{\alpha}{||\alpha||}
$$
The gauge group acts:
$$
f.(\alpha,\theta)=(f\alpha,\theta- df)
$$
Can we define invariants?