Orientifolds in Dimer Models: General Aspects and Applications.

In this thesis we study orientifolds and D-branes at toric Calabi-Yau singularities using dimer model techniques. We present new general results concerning orientifold projections and we apply them, constructing the first example of fully stable dynamical supersymmetry breaking model. The thesis is divided into three parts. In the first part we present mostly background material. We start with a short review of the Gauge/Gravity duality and generalizations. In particular we consider D3-branes at the tip of singular toric Calabi-Yau threefolds, introducing the idea of quiver theory and its natural generalization, dimer models. We then present known results obtained using this tool to explore orientifold projections. Finally, we review two dynamical supersymmetry breaking models, that are the simplest ones that can be engineered on D3-branes in this framework and which will play a prominent role in the last part of the thesis. The second part focuses on orientifolds. We present two new results: the first one links the geometry of the Calabi-Yau variety and the chiral anomaly of the orientifolded theory, in particular we will provide a criterium to determine whether an orientifold projection can be safely performed, i.e. without introducing uncancelled anomalies. The second concerns a new kind of orientifold projection related to glide involutions of the dimer model. This construction has the property to preserve superconformal invariance in the projected theory. In the third and last part of this thesis, we investigate dynamical supersymmetry breaking vacua obtained by D-branes at Calabi-Yau singularities. We first review known models and we show that all those theories have instabilities along $mathcal{N} = 2$ Coulomb branches. We then show that this instability can be overcome, with suitable generalizations of the models presented in the first part. Finally, using the results of the second part, we will show how to constrain the possible geometries able to host dynamical supersymmetry breaking models free of any instabilities. This will select a singular Calabi-Yau, that we dubbed the Octagon, which furnishes the first instance of a model able to host a fully stable dynamical supersymmetry breaking model within the Gauge/String correspondence.