Line bundle embeddings for heterotic theories

In heterotic string theories consistency requires the introduction of a non-trivial vector bundle. This bundle breaks the original ten-dimensional gauge groups ${\text{E}}_8\times {\text{E}}_8$ or $\text{SO}(32)$ for the supersymmetric heterotic string theories and $$\text{SO}$(16) \times $\text{SO}$(16)$ for the non-supersymmetric tachyon-free theory to smaller subgroups. A vast number of MSSM-like models have been constructed up to now, most of which describe the vector bundle as a sum of line bundles. However, there are several different ways of describing these line bundles and their embedding in the ten-dimensional gauge group. We recall and extend these different descriptions and explain how they can be translated into each other.