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We explore exact generalized symmetries in the standard 2+1d lattice\mathbb{Z}_2

ℤ_{2}

gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.

Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries

https://doi.org/10.1103/PhysRevLett.132.121601

Choi, Yichul; Forslund, Matthew; Lam, Ho Tat; Shao, Shu-Heng (March 2024, Physical Review Letters)

We derive model-independent quantization conditions on the axion couplings (sometimes known as the anomaly coefficients) to the standard model gauge group 1⁄2SUð3Þ × SUð2Þ × Uð1ÞY =Zq with q 1⁄4 1, 2, 3, 6. Using these quantization conditions, we prove that any QCD axion model to the right of the E=N 1⁄4 8=3 line on the jgaγγj-ma plot must necessarily face the axion domain wall problem in a postinflationary scenario. We further demonstrate the higher-group and noninvertible global symmetries in the standard model coupled to a single axion. These generalized global symmetries lead to universal bounds on the axion string tension and the monopole mass. If the axion were discovered in the future, our quantization conditions could be used to constrain the global form of the standard model gauge group.

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