Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds

We analyze heterotic line bundle models on elliptically fibered Calabi-Yau three-folds over weak Fano bases. In order to facilitate Wilson line breaking to the standard model group, we focus on elliptically fibered three-folds with a second section and a freely-acting involution. Specifically, we consider toric weak Fano surfaces as base manifolds and identify six such manifolds with the required properties. The requisite mathematical tools for the construction of line bundle models on these spaces, including the calculation of line bundle cohomology, are developed. A computer scan leads to more than $400$ line bundle models with the right number of families and an $SU(5)$ GUT group which can descend to standard-like models after taking the ${\mathbb{Z}}_2$ quotient. A common and surprising feature of these models is the presence of a large number of vector-like states.