Title:

Elliptic fibrations for Ftheory geometric engineering

String phenomenology aims to explain the physics of the universe in the context of string theory, the leading candidate to unify gravitational and quantum physics. A main ingredient in constructing such models is compactifying the tendimensional theory on a sixdimensional manifold, so that one is left with the four noncompact dimensions modelling spacetime. Performing this step allows one to reformulate many physical phenomena as properties of the geometry of the compactication manifold, and to nd generic constraints on physical models using methods from algebraic geometry. One such phenomenon in nature are gauge theories, both abelian and nonabelian. In this thesis, we undertake a systematic investigation of the interplay of the two in compactications of Ftheory. In Ftheory, the relevant compactication spaces are elliptically bered CalabiYau manifolds. They are particularly wellsuited to the study of gauge symmetries in string phenomenology, since they both allow the existence of exceptional gauge symmetries such as E6 as well as the localization of gauge degrees of freedom on subloci of the compactication manifold. Specically, nonabelian gauge theories are encoded as singularities of the elliptic bration, and the rational sections of the bration specify the abelian part of the gauge group. Using tools from algebraic geometry, we study singularities of elliptic brations with a group of rational sections of rank 1, i.e. with a single abelian gauge factor. In the rst part of the thesis, we rene the spectral cover formalism, which is a way to study local properties of the subloci of the compactication manifold on which gauge degrees of freedom are localized in Ftheory. We do so by introducing the spectral divisor. The spectral divisor allows one to construct gauge uxes in Ftheory in a purely local description. We exemplify this construction for an elliptic bration with associated gauge group E6. In the second part of the thesis, we use Tate's algorithm to obtain a comprehensive classication of singular bers and an explicit list of possible realizations of Ftheory compactications with both an abelian and nonabelian gauge symmetries. This list is complete for lowrank gauge symmetries, which are most relevant for building models of the universe, and thus allows to completely classify all Ftheory models with a single abelian gauge factor. In particular, this list includes phenomenologically interesting brations not considered in the literature before.
