Exploring the landscape of heterotic strings on T^d

TL;DR

The paper develops a lattice-embedding framework to classify gauge-group realizations in heterotic strings on T^d by embedding ADE root lattices into the Narain lattice II_{d,d+16}. It introduces several algorithms—EDD saturation, shift vectors, and neighborhood methods—to map moduli (Wilson lines, metric, B-field) to maximal gauge enhancements, and generalizes these constructions from S^1 to T^2 and beyond. By connecting to Nikulin’s lattice theory and Shimada–Shimada classifications, the work corroborates heterotic on T^2 duality with elliptic K3 fibrations and enumerates all maximal-rank groups for d=1 and d=2, including explicit moduli and complementary lattices T. The methods illuminate the structure of the Narain moduli space, reveal the role of the complementary lattice, and provide practical tools for exploring higher-dimensional toroidal compactifications, with implications for swampland criteria and heterotic/F-theory dualities.

Abstract

Compactifications of the heterotic string on T^d are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)^{d+16} left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II_{1,17}. More general algorithms to explore the moduli space are also developed. For d=1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in II_{d,d+16}. In agreement with the duality between heterotic on T^2 and F-theory on K3, all possible gauge groups on T^2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E_8 x E_8 and Spin(32)/Z_2 heterotic theories.

Figures (18)

• Figure 1: Extended Dynkin diagram for the $\mathrm{II}_{1,17}$ lattice, with labels showing the embedding of the extended Dynkin diagrams of ${\mathrm{E}}_8 + {\mathrm{E}}^{\prime}_8$. The Kac marks are shown in red.
• Figure 2: Extended Dynkin diagram for the $\mathrm{II}_{1,17}$ lattice, with labels showing the embedding of $\Gamma_{16}$. The Kac marks of the extended $\mathrm{SO}(32)$ diagram are shown in red.
• Figure 3: Simplest extended diagram for $T^2$ compactifications, reproducing the 44 maximal enhancements of $S^1$ compactifications times ${\mathrm{A}}_1$. All models have $A_2 = 0$ and $E = E_1$.
• Figure 4: Extended diagram with $A_2 = 0$ and $E = E_2$. This is the simplest example of an extended diagram with non-trivial new results.
• Figure 5: Extended diagram with $A_1= \delta_1 \times 0$, $A_2=0 \times \delta'_2$, $E=E_1$, which corresponds to the lattice $\text{II}_{1,9} + \text{II}_{1,9}$. Interestingly, these roots form a basis for $\text{II}_{2,18}$.