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

We consider Riemannian four-manifolds (X, gX) with a spin(c)-structure and a suitable circle bundle Y over X such that the spinc-structure on X lifts to a spin-structure on Y. With respect to these structures, a spinor phi on X lifts to an untwisted spinor psi on Y and a U(1)-gauge field A for the spin(c)-structure can be absorbed into a Kaluza-Klein metric g(Y)(A) on Y. We show that irreducible solutions (A, phi) to the Seiberg-Witten equations on (X, gX) for the given spin(c)-structure are equivalent to irreducible solutions. of a Dirac equation with cubic non-linearity on the Kaluza-Klein circle bundle (Y, g(Y)(A)). As an application, we consider solutions to the equations in the case of Sasaki five-manifolds, which are circle bundles over Kahler-Einstein surfaces.

Automated summary

This study focuses on Riemannian four-manifolds with spin(c)-structures and circle bundles, showing the equivalence between irreducible solutions to the Seiberg-Witten equations on the four-manifold and solutions to a Dirac equation on the Kaluza-Klein circle bundle. An application to Sasaki five-manifolds, circle bundles over Kahler-Einstein surfaces, is also considered.