Realizing higher-level gauge symmetries in string theory: New embeddings for string GUTs

We consider the methods by which higher-level and non-simply laced gauge symmetries can be realized in free-field heterotic string theory. We show that all such realizations have a common underlying feature, namely a dimensional truncation of the charge lattice, and we identify such dimensional truncations with certain irregular embeddings of higher-level and non-simply laced gauge groups within level-one simply laced gauge groups. This identification allows us to formulate a direct mapping between a given subgroup embedding, and the sorts of GSO constraints that are necessary in order to realize the embedding in string theory. This also allows us to determine a number of useful constraints that generally affect string GUT model-building. For example, most string GUT realizations of higher-level gauge symmetries G_k employ the so-called diagonal embeddings G_k ⊂ G × G × ⋯ × G. We find that there exist interesting alternative embeddings by which such groups can be realized at higher levels, and we derive a complete list of all possibilities for the GUT groups SU(5), SU(6), SO(10), and E₆ at levels k = 2, 3, 4 (and in some cases up to k = 7). We find that these new embeddings are always more efficient and require less central charge than the diagonal embeddings which have traditionally been employed. As a by-product, we also prove that it is impossible to realize SO(10) at levels k > 4 in string theory. This implies, in particular, that free-field heterotic string models can never give a massless 126 representation of SO(10).