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 E₈×E₈ or SO(32) for the supersymmetric heterotic string theories and SO(16)×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.