A lift of the Seiberg-Witten equations to Kaluza-Klein 5-manifolds

Abstract

We consider Riemannian 4-manifolds X with a Spin c -structure and a suitable circle bundle Y over X such that the Spin c -structure on X lifts to a spin structure on Y . With respect to these structures a spinor on X lifts to an untwisted spinor on Y and a U(1)-gauge field for the Spin c -structure can be absorbed in a Kaluza-Klein metric on Y . We show that irreducible solutions to the Seiberg-Witten equations on X are equivalent to solutions of a Dirac equation for the spinor on Y with a cubic non-linearity.