## Abstract

Pseudo-Riemannian spinc 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 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

## Abstract

In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle *T M* ⊕ *T* M* that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle *E* ⊕ *E**, where E is a Banach Lie algebroid and *E** its dual. Recall that *E** is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle *E* ⊕ *E** and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.