Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

up vote 5 down vote accepted

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.)

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.