Holomorphic Yukawa Couplings in Heterotic String Theory

TL;DR

This thesis develops a method to compute holomorphic Yukawa couplings in heterotic string models on Calabi-Yau manifolds, enabling explicit calculations and analysis of their dependence on moduli and topology.

Contribution

It introduces a novel technique relating bundle forms to ambient space forms for Yukawa coupling calculations in heterotic models with line bundle sums.

Findings

Explicit Yukawa couplings computed for specific CICYs.

Topology can cause the absence of certain Yukawa couplings.

Yukawa matrices depend on complex structure moduli and can have reduced rank.

Abstract

This thesis is concerned with heterotic E8 x E8 string models that can produce quasirealistic N = 1 supersymmetric extensions of the Standard Model in the low-energy limit. We start rather generally by deriving the four-dimensional spectrum and Lagrangian terms from the ten-dimensional theory, through a process of compactification over six-dimensional Calabi-Yau manifolds, upon which holomorphic poly-stable vector bundles are defined. We then specialise to a class of heterotic string models for which the vector bundle is split into a sum of line bundles and the Calabi-Yau manifold is defined as a complete intersection in projective ambient spaces. We develop a method for calculating holomorphic Yukawa couplings for such models, by relating bundle-valued forms on the Calabi-Yau manifold to their ambient space counterparts, so that the relevant integrals can be evaluated over projective spaces. The method is applicable for any of the 7890 CICY manifolds known in the literature, and we show that it can be related to earlier algebraic techniques to compute holomorphic Yukawa couplings. We provide explicit calculations of the holomorphic Yukawa couplings for models compactified on the tetra-quadric and on a co-dimension two CICY. A vanishing theorem is formulated, showing that in some cases, topology rather than symmetry is responsible for the absence of certain trilinear couplings. In addition, some Yukawa matrices are found to be dependent on the complex structure moduli and their rank is reduced in certain regions of the moduli space. In the final part, we focus on a method to evaluate the matter field Kahler potential without knowing the Ricci-flat Calabi-Yau metric. This is possible for large internal gauge fluxes, for which the normalisation integral localises around a point on the compactification manifold.