On the Gauge Group Topology of 8d CHL Vacua

TL;DR

This work develops a lattice-based framework to determine the full gauge-group topology, including U(1) factors, in 8d CHL vacua by embedding root lattices into the Mikhailov momentum lattice $\Lambda_M$ and reading the global structure from the dual lattice $\Lambda_M^*$. It provides explicit prescriptions for extracting $\pi_1(G)$ and the center constraints $\mathcal Z'$ from lattice data, and proves that CHL gauge groups are free of a mixed 1-form anomaly, with explicit confirmation for rank-10 maximally enhanced vacua. The authors illustrate the method with a concrete CHL example and show how, via a projection map, CHL data can be derived from a parent heterotic model, establishing a practical CHL-heterotic correspondence. The results generalize our understanding of gauge-group topology in CHL theories and connect to F-theory via discriminant forms, string junctions, and higher-form symmetry considerations, offering a robust toolkit for classifying 8d gauge topologies and guiding extensions to broader compactifications.

Abstract

Compactifications of the CHL string to eight dimensions can be characterized by embeddings of root lattices into the rank 12 momentum lattice $Λ_M$, the so-called Mikhailov lattice. Based on this 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 non-trivial $π_1(G) \equiv {\cal Z}$ for the non-Abelian gauge group $G$ as having gauged a ${\cal Z}$ 1-form symmetry, we also prove that all CHL gauge groups are free of a certain anomaly (arXiv:2008.10605) that would obstruct this gauging. We verify this by explicitly computing ${\cal Z}$ for all 8d CHL vacua with 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.