A Note on the Geometry of CHL Heterotic Strings

TL;DR

This work shows that disconnected components of the heterotic moduli space on a two-torus, notably the CHL and Narain sectors, have a unified geometric origin in flat $G$-bundles with non-simply connected structure groups. By analyzing flat connections on elliptic curves and employing folding of affine Dynkin diagrams, the authors demonstrate that topologically non-trivial sectors of $G$-bundles are isomorphic to trivial sectors with folded structure groups $G^\omega$, and illustrate this explicitly for $G={\rm Spin}(32)/\mathbb{Z}_2$, where the CHL component corresponds to the trivial Spin$(17)$ moduli. They then realize the eight-dimensional CHL compactification by choosing specific Wilson lines and boundary conditions that interchange the two $E_8$ factors on one cycle, establishing the equivalence of the CHL moduli with the corresponding non-trivial Spin$(32)/\mathbb{Z}_2$ sector and expressing the full moduli as $\mathcal{M}_{18,2} \sim \mathcal{M}_{\mathrm{Spin}(32)} \times \mathcal{M}_{2,2}$ and $\mathcal{M}_{10,2} \sim \mathcal{M}_{\mathrm{Spin}(17)} \times \mathcal{M}_{2,2}/\mathbb{Z}_2$. The analysis highlights a dual-gauge-group perspective, where the natural gauge group is the Langlands dual $G^{\vee}$, and points toward a future F-theory dual description of CHL. These results provide a concrete geometric framework for CHL and clarify its relationship to Narain compactifications.

Abstract

We present a few remarks on disconnected components of the moduli space of heterotic string compactifications on $T_2$. We show in particular how the eight dimensional CHL heterotic string can be understood in terms of topologically non-trivial $E_8\times E_8$ and $\Spin(32)/Z_2$ vector bundles over the torus, and that the respective moduli spaces coincide.

Figures (1)

• Figure 1: Some foldings of affine Dynkin diagrams.