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

TL;DR

<3-5 sentence high-level summary>This paper provides a unified framework for realizing higher-level and non-simply laced gauge symmetries in free-field heterotic string theories by identifying dimensional truncations of the charge lattice as the core mechanism. It develops a general formalism linking irregular subgroup embeddings to specific GSO projections through an embedding matrix and its nullspace, enabling explicit construction of higher-level GUTs beyond the traditional diagonal embeddings. The authors classify all viable embeddings for SU(5), SU(6), SO(10), and E6 at levels $k=2,3,4$ (and SO(10) up to $k=7$), proving, in particular, that SO(10) cannot be realized for $k>4$ and that SO(10) massless ${\bf 126}$ states are forbidden in free-field models. They also demonstrate that certain non-diagonal embeddings are more efficient (less central charge) than diagonal ones, with concrete GSO projections provided for key cases like $SO(10)_2$ and $SU(5)_3$ from $SU(10)_1$. These results have direct implications for string GUT model-building and the allowed matter content in realistic constructions.

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).

Figures (5)

• Figure 1: Procedures for string model-building, as discussed in the text. Particular choices for the free parameters within a given string construction determine the resulting GSO projections, which in turn define a particular string model. The desired phenomenology, on the other hand, dictates a specific group-theoretic embedding of the gauge and matter representations. The connection between the construction procedure and the desired group-theoretic embedding occurs through the charge lattice, at the level of the GSO projections.
• Figure 2: The root system of $SU(2)_1^{(A)}\times SU(2)_1^{(B)}$ (denoted by open circles), and its dimensional truncation onto the diagonal subgroup $SU(2)_2^{(V)}$ (with new non-zero roots denoted by shaded circles).
• Figure 3: The irregular embedding of $SU(2)_4$ within $SU(3)_1$. The non-zero roots of $SU(3)_1$ lie along the outer hexagon, as denoted by empty circles. The roots labelled $\vec{\alpha}_1$ and $\vec{\alpha}_2$ are the simple roots, and the weights of the fundamental 3 representation of $SU(3)$ are also shown (shaded circles). Next to each weight we have listed its Dynkin labels, along with the Dynkin label of the $SU(2)$ weight to which it corresponds according to the embedding matrix in Eq. (\ref{['Psu3su2']}). The axis of $SU(2)$ truncation is also shown (dark line), with the vector $\vec{\beta}$ (black square) chosen perpendicular to this axis.
• Figure 4: The irregular embedding of $SU(2)_{10}$ within $SO(5)_1$. The non-zero roots of $SO(5)_1$ lie along the outer square, with the empty circles denoting the long roots and the black circles denoting the short roots. The simple roots are $\vec{\alpha}_1$ and $\vec{\alpha}_2$. The non-zero weights of the 5 representation of $SO(5)$ comprise the short roots alone, and the weights of the 4 representation are also superimposed (shaded circles). Next to each weight we have listed its Dynkin labels, along with the Dynkin label of the $SU(2)$ weight to which it corresponds according to the embedding matrix in Eq. (\ref{['Pso5su2']}). The axis of $SU(2)$ projection is also shown (dark line), with the vector $\vec{\beta}$ (black square) chosen perpendicular to this axis. The Dynkin labels of $\vec{\beta}$ thus uniquely define this axis, and specify the orientation angle to be $\theta=\cos^{-1}(2/\sqrt{5})$.
• Figure 5: Distribution of rank and central charge for the general embedding of $G'_{k'}$ within a level-one simply laced group $G_1$. Here $\Delta r$ and $\Delta c$ are the total rank and central-charge "costs" of realizing the embedding. The degrees of freedom represented by $\Delta r_2=\Delta c$, in particular, form a (usually unavoidable) independent extra chiral algebra which does not contribute to the gauge symmetries of the theory.