Unoriented dualities in mass deformed toric gauge theories

This thesis focuses on a certain class of supersymmetric gauge theories. In particular, we study the interplay between relevant mass deformations of these theories and a projection known as orientifold. Using the usual tools for $mathcal{N}=1$ theories, such as anomaly matching and $a$-maximization, we look for the conditions for the theories to be conformally invariant after the projection and classify the possible outcomes into three scenarios. We find that for certain families of models the mass deformation is no longer relevant, as symmetries constrain this to be exactly marginal. Moreover, we find a web of dualities that connects these projected models and construct infinitely many non-chiral examples, along with a single dual chiral pair. The theories of interest arise as the gauge side of the AdS/CFT correspondence, while their gravity side is defined over the so-called toric geometries. Thus, projection and deformation have a geometric counterpart in the gravity theory. In this context, we use tools as brane tiling to construct our models. However, the geometric interpretation associated with our new found web of dualities is not completely understood.