Pseudo-Riemannian spinc manifolds were introduced by Ikemakhen in . 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 spinc 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 at space
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 Matsushita Y., The existence of indefinite metrics of signature (+;+;−;−) and two kinds of almost complex structures in dimensionFour, Proceedings of The Seventh International Workshop on Complex Structures and Vector Fields, Contemporary Aspects of Complex Analysis, Differential Geometry and Mathematical Physics, ed. S. Dimiev and K. Sekigawa, World Scientific, 210-225, 2005.
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