Supersymmetric Localization : A Journey from Seven to Three Dimensions

Quantum Field Theory has been a dominating framework in elementary particle physics during the last century. Within this framework, supersymmetric theories have attracted a lot of attention due to their mathematical structure, simplicity and insight into the problems of unification, dark matter and hierarchy. Even though the boundaries of our perturbative undestanding of supersymmetric theories have been pushed far, there is generically no systematic way to obtain exact results. Especially for strongly coupled theories, where perturbative techniques cannot be applied, methods for exact computations are crucial. A powerful technique to obtain exact results of partition functions and correlators of curtain protected operators is supersymmetric localization. This thesis studies physical and geometrical properties of localization results in different supersymmetric theories. The first model considered is maximally supersymmetric Yang-Mills placed on a 7-dimensional Sasaki-Einstein manifold. After redefining the fields to differential forms, the cohomological complex is formed and a localization computation of the partition function is performed using data from the moment map cone. This procedure and the factorization properties reveal a strong structural dependence of the result on the geometry. A second part discusses N=4 matter-multiplets in 3d. We identify BPS operators supported on a 1-dimensional submanifold. Applying the localization formula, the partition function simplifies to a partition function of a dual topological quantum mechanics. Lastly, we perform an equivariant twisting on N=2 gauge theories with matter on 4-dimensional manifolds with a torus action. This is achieved by a global field redefinition leading to differential forms or spinors that are defined on a large class of manifolds. The resulting cohomological theory admits two kinds of fixed points which are treated differently in a localization computation