Abstract
Compactifications of the Chaudhuri-Hockney-Lykken (CHL) string to eight dimensions can be characterized by embeddings of root lattices into the rank 12 momentum lattice ${\mathrm{\ensuremath{\Lambda}}}_{M}$, the so-called Mikhailov lattice. Based on these data, we devise a method to determine the global gauge group structure including all $U(1)$ factors. The key observation is that, while the physical states correspond to vectors in the momentum lattice, the gauge group topology is encoded in its dual. Interpreting a nontrivial ${\ensuremath{\pi}}_{1}(G)\ensuremath{\equiv}\mathcal{Z}$ for the non-Abelian gauge group $G$ as having gauged a $\mathcal{Z}$ 1-form symmetry, we also prove that all CHL gauge groups are free of a certain anomaly [1] that would obstruct this gauging. We verify this by explicitly computing $\mathcal{Z}$ for all 8D CHL vacua with $\mathrm{rank}(G)=10$. Since our method applies also to ${T}^{2}$ compactifications of heterotic strings, we further establish a map that determines any CHL gauge group topology from that of a ``parent'' heterotic model.
Highlights
Supersymmetric string compactifications on lowdimensional internal manifolds have seen a resurgence of interest within the Swampland program [2,3]
The key observation is that, while the physical states correspond to vectors in the momentum lattice, the gauge group topology is encoded in its dual
Interpreting a nontrivial π1ðGÞ ≡ Z for the non-Abelian gauge group G as having gauged a Z 1-form symmetry, we prove that all CHL gauge groups are free of a certain anomaly [1] that would obstruct this gauging
Summary
Supersymmetric string compactifications on lowdimensional internal manifolds have seen a resurgence of interest within the Swampland program [2,3]. Any non-Abelian gauge algebra g that can arise in an 8D CHL vacuum must have a root lattice Λgr that embeds in a specific way into ΛM Such lattice embeddings can be classified [16] in an analogous fashion as for rank 20 theories based on their heterotic realization [12], where the corresponding string momentum lattice is the rank 20 Narain lattice ΛN [17,18]. This is the dual of the character lattice ΛGc , which corresponds to the charge lattice occupied by physical states, which clearly is the momentum lattice ΛS From this perspective, the self-duality of the Narain lattice (imposed by modularity of the heterotic world sheet), together with the fact that rank 20 theories only have ADE algebras [whose (co-)root lattices Λgr 1⁄4 Λgcr agree], appear as a coincidence that makes it straightforward to compute the fundamental group Z 1⁄4 π1ðGÞ as (the torsional piece of) ΛN=Λgr , as done in [12].
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