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We derive model-independent quantization conditions on the axion couplings(sometimes known as the anomaly coefficients) to the Standard Model gauge group$[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_q$ with $q=1,2,3,6$. Using thesequantization conditions, we prove that any QCD axion model to the right of the$E/N=8/3$ line on the $g_{a\gamma\gamma}$-$m_a$ plot must necessarily face theaxion domain wall problem in a post-inflationary scenario. We furtherdemonstrate the higher-group and non-invertible global symmetries in theStandard Model coupled to a single axion. These generalized global symmetrieslead to universal bounds on the axion string tension and the monopole mass. Ifaxion were discovered in the future, our quantization conditions could be usedto constrain the global form of the Standard Model gauge group.

Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries