Abstract |
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In 1954, C. Yang and R. Mills created a Gauge Theory for strong interaction of Elementary Particles. More generally, they proved that it is possible to define a Gauge Theory with an arbitrary compact Lie group as Gauge group. Within this context, it is interesting to find critical values of a functional defined on the space of connections: the Yang-Mills functional. If the based manifold is four dimensional, there exists a natural notion of (anti-)self-dual 2-form, which gives a natural notion of (anti-)self-dual connection. Such connections give critical values of the Yang-Mills functional. Moreover, the Gauge group acts on the set of (anti-)self-dual connections. The set of (anti-)self-dual connections modulo the Gauge group is called the Moduli space of (anti-)self-dual connections. It is interesting for physicists because it provides critical values of the Yang-Mills functional and for mathematicians because it is an invariant of the based manifold. In dimension greater than four, it is possible to extend this notion of (anti-)self-duality in different suitable ways. We are working on almost Kähler manifolds – so in particular on every symplectic manifolds. On almost Kähler manifolds, an extension of self-duality appears naturally. In Chapter 2, we define this extended notion of self-duality of 2-forms and we determine the space of self-dual 2-forms. In Chapter 3, we begin by explaining how this definition of self-duality of 2-forms gives a definition of self-duality of connections. Then, we define the corresponding Moduli space of self-dual connections. Finally, we identify suitable hypotheses under which we are able to characterize the moduli space of self-dual connections and to build a Lie group structure on it. |