Title:

Heterotic string models on smooth CalabiYau threefolds

This thesis contributes with a number of topics to the subject of string compactifications, especially in the instance of the E_{8} × E_{8} heterotic string theory compactified on smooth CalabiYau threefolds. In the first half of the work, I discuss the Hodge plot associated with CalabiYau threefolds that are hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of ellipticK3 fibrations, whose mirror images are also ellipticK3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fiber. Any two halfpolytopes 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 of manifolds, by looking at smooth Z_{3}quotients of CalabiYau threefolds realised as complete intersections in products of projective spaces. Nonsimply connected CalabiYau threefolds provide an essential ingredient in heterotic string compactifications. Such manifolds are rare in the classical constructions, but they can be obtained as quotients of homotopically trivial CalabiYau threefolds by free actions of finite groups. Many of these quotients are connected by conifold transitions. In the second half of the work, I explore an algorithmic approach to constructing E_{8} × E_{8} heterotic compactifications using holomorphic and polystable sums of line bundles over complete intersection CalabiYau threefolds that admit freely acting discrete symmetries. Such Abelian bundles lead to N = 1 supersymmetric GUT theories with gauge group SU(5) × U(4) and matter fields in the 10, ⁻10, ⁻5, 5 and 1 representations of SU(5). The extra U(1) symmetries are generically GreenSchwarz anomalous and, as such, they survive in the low energy theory only as global symmetries. These, in turn, constrain the low energy theory and in many cases forbid the existence of undesired operators, such as dimension four or five proton decay operators. The line bundle construction allows for a systematic computer search resulting in a plethora of models with the exact matter spectrum of the Minimally Supersymmetric Standard Model, 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 a CalabiYau hypersurface embedded in a product of four CP1 spaces, referred to as the tetraquadric manifold. I address the question of the finiteness of the class of consistent and physically viable line bundle models constructed on this manifold. Line bundle sums are part of a moduli space of nonAbelian bundles and they provide an accessible window into this moduli space. I explore the moduli space of heterotic compactifications on the tetraquadric hypersurface around a locus where the vector bundle splits as a direct sum of line bundles, using the monad construction. The monad construction provides a description of polystable S(U(4) × U(1))–bundles leading to GUT models with the correct field content in order to induce standardlike models. These deformations represent a class of consistent nonAbelian models that has codimension one in Kähler moduli space.
