I define equations like the Seiberg-Witten equations for forms of a riemannian four-manifold $(M,g)$. $\alpha \in \Lambda^2 (TM)$ 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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7$\begingroup$ 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 GerigCommented Mar 22, 2020 at 16:54
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1$\begingroup$ But there is a real analogy between these equations and those of Seiberg-Witten. The question deserves to be posed. $\endgroup$– Antoine BalanCommented Mar 22, 2020 at 18:31
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$\begingroup$ trivial remark: if we think of $\theta$ as associated to a connection, then the first equation just says that $\alpha$ is a parallel $2$-form with respect to that connection. The second equation seems to say something about the curvature of this connection. $\endgroup$– MalkounCommented Jan 11 at 20:14
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