Localization Techniques, Yang-Mills Theory and Strings

Sammanfattning: Equivariant localization techniques exploit symmetries of systems, represented by group actions on manifolds, and use them to evaluate certain partition functions exactly. In this master thesis we begin with the study of localization in finite dimensions. We then generalize this concept to infinite dimensions and study the partition function of two dimensional quantum Yang- Mills theory and its relation to string theory. The partition function can be written as a sum over the critical point set and be related to the topology of the moduli space of flat connections. Furthermore, for large N the partition function of the gauge groups SU(N) and U(N) can be interpreted as a string perturbation series. The coefficients of the expansion are given by a sum over maps from a two dimensional surface onto the two dimensional target space and thus the partition function is interpreted as a closed string theory. Also, a string theory action is discussed using topological field theory tools and localization techniques.