What are the irreducible solutions of SeibergWitten equation on S^2\times S^1? Thanks.
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The equations depends on a Riemannian metric and on a perturbation term (a closed 2form). The usual metric on $S^2\times S^1$ has positive scalar curvature, which implies that there are no irreducible solutions when the 2form vanishes. This is a consequence of the Weitzenboeck formula for the Dirac operator, as was observed by Witten in his paper Monopoles and fourmanifolds. (The 3dimensional SW equations are the 4dimensional SW equations on the product with $S^1$, with the additional condition of translationinvariance in the circledirection.) 

