I define equations like the Seiberg-Witten equations for forms of a riemannian four-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:
$$
f.(\alpha,\theta)=(f\alpha,\theta- df/f)
$$
Can we define invariants?