Localization of N=2 Field Theories: a Twisted Path Across Four Manifolds

Supersymmetric quantum field theories provide a framework where certain physical observables can be computed exactly. In those cases, one not only has control over perturbative contributions but also over non-perturbative contributions. In this thesis the main focus are N=2 supersymmetric quantum field theories on compact manifolds with U(1)xU(1) isometry and a Killing vector with isolated fixed points. In Part I, focusing on pure gauge theories, it is explained how equivariant Donaldson-Witten theory and a certain class of non-topological theories, related to the well-known result of Pestun on the four-dimensional sphere, can be described as two instances of an underlying framework. Employing this formalism, a general formula for the partition functions has been proposed which is valid both for equivariant Donaldson-Witten and Pestun-like theories. On top of perturbative contributions, the partition functions get contributions from instantons and fluxes. In Part II, the results appearing in the papers attached to this thesis are presented. First, a formal treatment of the perturbative part is discussed. Then, the dependence on flux of the partition function is studied and it is shown how Donaldson-Witten and Pestun-like theories arise from a unique five-dimensional theory, after dimensional reduction. Finally, matter coupled to gauge fields are included in the framework above