Solving N = 2 gauge theories through matrix models on the sphere

In this thesis, we investigate various classes of four-dimensional N=2 super–Yang–Mills (SYM) theories, including both conformal and non-conformal setups. The extended but non-maximal supersymmetry of these models makes them an ideal framework for exploring quantum field theory beyond perturbation theory through non-perturbative approaches such as supersymmetric localization. We study the parameter space of N=2 superconformal quiver theories by computing protected correlation functions that capture the dynamics of the underlying models. Using the matrix model derived from localization on the four-sphere, we obtain exact results at weak and strong coupling, revealing the emergence of effective five-dimensional dynamics in the deconstruction limit. We then extend these matrix-model techniques to non-conformal N=2 theories, focusing on supersymmetric Wilson loops and chiral correlators. By comparing localization results on the four-sphere with perturbative computations in flat space, we demonstrate a precise agreement within a specific regime of validity, thereby broadening the applicability of localization methods beyond conformal settings.