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