Results in Localization for Supersymmetric Gauge Theories

The strong coupling dynamics of Quantum Field Theories with gauge symmetries constitutes a profound problem in Theoretical Physics. Supersymmetric theories offer rare instances where this elusive problem is tractable and can be a valuable source of information and intuition. One of the most powerful approaches available for this purpose is supersymmetric localization. In this thesis, we employ this technique to study many facets of supersymmetric theories. After a brief theoretical introduction to the basic concepts that make an appearance in this thesis, we begin with a localization computation for supersymmetric gauges theories on spheres of variable dimension, with eight and four supersymmetries, proving an earlier conjecture. We also analytically continue our results to get a partition function for the four-dimensional N=1 theory, and perform various consistency checks. Then, we consider a special case of these theories, namely two-dimensional Yang-Mills gauge theory, and we study the matrix model that arises from its localization. We analyze the theory in the limit of large gauge group rank using numerical and analytical tools and connect it to previous related results from the literature. Next, we turn our attention to N=2 gauge theories with matter. We construct the theories on a general class of four-dimensional manifolds, ensuring that they are globally well-defined. Then, we reformulate them using twisted variables and perform localization on the resulting cohomological theory. Lastly, we study N=2 theories on another four-dimensional manifold, namely a product of a three-dimensional hyperbolic space and a circle, with the goal of finding infinite-dimensional chiral algebras. To pursue this goal, we once again employ a localization technique and find evidence of this structure in two cases