Heterotic String Models on Smooth Calabi-Yau Threefolds

Abstract

This thesis contributes with a number of topics to the subject of string compactifications. In the first half of the work, I discuss the Hodge plot of Calabi-Yau threefolds realised as hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along K3 slices. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, give to the Hodge plot the appearance of a fractal. Moving on, I discuss a different type of web by looking at smooth $\mathbb{Z}_3$-quotients of complete intersection Calabi-Yau threefolds in products of projective spaces. In the second half of the work, I explore an algorithmic approach to constructing heterotic compactifications with holomorphic and polystable sums of line bundles over complete intersection Calabi-Yau threefolds that admit freely acting discrete symmetries. Such abelian bundles lead to $N=1$ GUT theories with gauge group $SU(5)\times U(4)$. The extra $U(1)$ symmetries are generically Green-Schwarz anomalous and survive in the low energy theory as global symmetries that constrain the allowed operators. The line bundle construction allows for a systematic computer search resulting in a plethora of models with one or more pairs of Higgs doublets and no exotic fields charged under the Standard Model group. In the last part of the thesis I focus on the case study of the tetraquadric threefold and address the question of the finiteness of the class of consistent and physically viable line bundle models on this manifold. Line bundle sums provide an accessible window into the moduli space of non-abelian bundles. I explore this moduli space around the abelian locus using monad bundles.