On path integral localization and the Laplacian

Sammanfattning: In this thesis, we develop path integral localization methods that are familiar from topological field theory: the integral over the infinite dimensional integration domain depends only on local data around some finite dimensional subdomain.We introduce a new localization principle that unifies BRST localization, the non-Abelian localization principle and the conformal generalization of the Duistermaat-Heckman integration formula..In addition, it is studied if one can possibly derive a generalized Selberg's trace formula on locally homogeneous manifolds using localization techniques. However, a definite answer is obtained only in the Lie group case (we complete the work of R. Picken) in which it is an application of the Duistermaat-Heckman integration formula. Also a new derivation of DeWitt's term is reported.Furthermore, connections between evolution operators of integrable models and localization methods are studied. A derivative expansion localization is presented and it is conjectured to apply also to integrable models, for example the Toda lattice.Moreover, a pedagogical introduction to the localization techniques is given, as well as a list of selected references that might be useful for a beginning graduate student in mathematical physics or for a mathematician who would like to study the physical point of view to topological field theory and string theory.