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For a vector field $X$ and a spinor $\psi$, we can define the modified Seiberg-Witten equations:

$$D_X \psi=(D+i X) \psi=0$$

$$i d(X^*)_+ = -(1/4)w(\psi )/|\psi |^2 $$

with $D$ the Dirac operator, $X^*$ is the dual 1-form of $X$; $+$ is the self-dual part and $w$ is the 2-form bound to the spinor $\psi$.

The gauge group acts: $f.(X,\psi )=(X- (d f)^*, if \psi)$

Have we compact moduli spaces?

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