# Seiberg-Witten equation on S^2\times S^1

What are the irreducible solutions of Seiberg-Witten equation on S^2\times S^1? Thanks.

-

## 1 Answer

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.)

-
OK. Thanks. So it is not a topological invariant? –  Xuanting Cai Jan 27 '11 at 23:19
No: the topological invariant is the fundamental class of the SW moduli space (for metrics and perturbations with generic properties, including the absence of reducible solutions) in the homology of the the space of irreducible configurations mod gauge. –  Tim Perutz Jan 28 '11 at 3:29