Seiberg-Witten Equations on Pseudo-Riemannian Spin(c) Manifolds With Neutral Signature

Abstract

Pseudo-Riemannian spin(c) manifolds were introduced by Ikemakhen in [7]. In the present work we consider pseudo-Riemannian 4-manifolds with neutral signature whose structure groups are SO+ (2, 2). We prove that such manifolds have pseudo-Riemannian spin(c) structure. We construct spinor bundle S and half-spinor bundles S+ and S- on these manifolds. For the first Seiberg-Witten equation we define Dirac operator on these bundles. Due to the neutral metric self-duality of a 2-form is meaningful and it enables us to write down second Seiberg-Witten equation. Lastly we write down the explicit forms of these equations on 4-dimensional flat space.