On abelian and discrete symmetries in F-theory

In this dissertation, we systematically construct and study global F-theory compactifications with abelian and discrete gauge groups. These constructions are of fundamental relevance for both conceptual and phenomenological reasons. In the case of abelian symmetries, we systematically engineer compactifications that support U(1)×U(1) and U(1)×U(1)×U(1) gauge groups. The engineered geometries are elliptic fibrations with Mordell-Weil group rank two and three respectively. The bases of the fibrations are arbitrary, but as proofs of concept, we explicit create examples with bases P 2 and P3. We study the low energy physics of these compactifications, we calculate the matter spectrum and confirm that it is anomaly free. In 4D compactifications, the G4 flux is designed and the existence of Yukawa couplings is verified. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. The discrete gauge groups Z3 and U(1) × Z2 are naturally found when P2 and P1 × P1 are used as fiber ambient spaces. We also find the first realization of matter with U(1) charge three. Finally, we study the discrete gauge group Z 3 in detail. We find the three elements of the Tate-Shafarevich (TS) group. We make use of the Higgs mechanism with the charge three hypermultiplets and the Kaluza-Klein reduction from 6D to 5D. The results are interpreted from the F- M- theory duality perspective. In F-theory, compactifications over any of the three elements of the TS groups yield the same low energy physics, however, M-theory compactifications over the same elements give rise to different gauge groups