# The Seiberg-Witten equations for forms

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?

• 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.). – Chris Gerig Mar 22 at 16:54
• But there is a real analogy between these equations and those of Seiberg-Witten. The question deserves to be posed. – Antoine Balan Mar 22 at 18:31