Gauge group topology of 8D Chaudhuri-Hockney-Lykken vacua

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 <math display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>M</mi></msub></math>, the so-called Mikhailov lattice. Based on these data, we devise a method to determine the global gauge group structure including all <math display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> 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 <math display="inline"><msub><mi>π</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≡</mo><mi mathvariant="script">Z</mi></math> for the non-Abelian gauge group <math display="inline"><mi>G</mi></math> as having gauged a <math display="inline"><mi mathvariant="script">Z</mi></math> 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 <math display="inline"><mi mathvariant="script">Z</mi></math> for all 8D CHL vacua with <math display="inline"><mrow><mi>rank</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mn>10</mn></mrow></math>. Since our method applies also to <math display="inline"><msup><mi>T</mi><mn>2</mn></msup></math> compactifications of heterotic strings, we further establish a map that determines any CHL gauge group topology from that of a “parent” heterotic model.