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?

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    $\begingroup$ You're asking a big question (almost homework-like), like "show me how to build a cohomology theory", without demonstrating any attempt yourself, which is why I voted to close. I recommend understanding how the ordinary Seiberg-Witten invariants are defined, and check it for yourself (do you have an elliptic equation, transversality, appropriate index, compactness, etc.). $\endgroup$ – Chris Gerig Mar 22 at 16:54
  • $\begingroup$ But there is a real analogy between these equations and those of Seiberg-Witten. The question deserves to be posed. $\endgroup$ – Antoine Balan Mar 22 at 18:31

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