Realizing Higher-Level Gauge Symmetries in String Theory: New Embeddings for String GUTs

TL;DR

This paper explores new methods for embedding higher-level and non-simply laced gauge symmetries in heterotic string theory, revealing more efficient embeddings for string GUTs and establishing constraints on realizations of certain gauge groups.

Contribution

It introduces a unified framework linking charge lattice truncations to irregular embeddings, and provides a complete classification of embeddings for key GUT groups at various levels.

Findings

Identifies common features of higher-level gauge symmetry realizations

Derives a complete list of alternative embeddings for GUT groups at levels 2-7

Proves the impossibility of realizing SO(10) at levels above 4

Abstract

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\subset G\times G \times...\times 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 E_6 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 byproduct, we also prove that it is impossible to realize SO(10) at levels k>4. This implies, in particular, that free-field heterotic string models can never give a massless 126 representation of SO(10).