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.