Generalizations of Seiberg-Witten equations

In this thesis, Seiberg-Witten equations have been examined under two main catagories. Firstly, some basic concepts such as Clifford algebra, Vector bundles, Principal bundles and Connection 1-form were addressed to describe Seiberg-Witten equations. In the next section, after Dirac operator and covariant derivative operator were studied on spinor bundle, Seiberg-Witten equations, which are used to analyze the structure of the 4-manifold and well-$known in the literature, have been discussed. According to this, Seiberg-Witten equations have been written on the 4-dimensional Hyperbolic space. In higher dimension, depending on the concept of the generalized self-duality, alternative formulas of Curvature equation, which are equivalent to wording in the literature, was given. In 8-dimension Seiberg-Witten equations were also obtained according to the different selection of a self-duality and then solutions to these equations were given. At the end of this section, Seiberg-Witten equations on 8-dimensional Hyperbolic space were written. Then, some properties of the quadratic map sigma, which is used in the expressions of the Curvature equation, were investigated on the spinor space by using defined hermitian inner product. According to these, some useful equations were obtained. In the final section, instead of clasical equations which are defined on 4-dimensional manifolds, at first an alternative approach has been suggested without the need of self-duality concept. Finally, by this approach, without using the concept of self-duality Seiberg-Witten equations were written and solutions to these equations were given in dimensions 5,6,7 and 8. Keywords: Dirac Operator, Seiberg-Witten Equations, Spinor, Curvature, Self-Duality