Applications of Seiberg-Witten Equations to the Topology of Smooth Manifolds
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Department of Mathematics, Faculty of Sciences and Arts, Ağrı İbrahim Çeçen University,
Seiberg−Witten theory has played an important role in the topology of 4−manifolds. Seiberg−Witten equations consists of two equations, curvature equation and Dirac equation. Dirac equation can be written down on any Spinc manifold of any dimension. Due to the self−duality of a 2−form, the curvature equation is special to 4−dimensional manifolds. There are some generalizations of these equations to higher dimensional manifolds. All of them are mainly based on the generalized self−duality of a 2−form . Since any almost hermitian manifold has a canonical Spinc−structure which is determined by its almost hermitian structure, a fundemantal role is played by the almost hermitian manifold. As in almost hermitian manifolds, every contact metric manifold can be equipped with the canonical Spinc−structure determined by the almost complex structure on the contact distribution. Also, on these manifolds, one can described a spinor bundle. Therefore, for a given canonical Spinc−structure on the contact metric manifold, a spinorial connection can be deﬁned on the associated spinor bundle by means of the generalized Tanaka−Webster connection . The Dirac operator is associated to a such connection. The curvature equation which couples the self−dual part of the curvature 2−form with a spinor ﬁeld.
Aims and Scope Defining Seiberg-Witten equation on n-dimensional manifold Giving global solution to these equations Obtaining differential topological invariants for smooth-manifolds via the solution space of these equations