Lazarides-Shafi axion models as Dijkgraaf-Witten theories

Axion models often face the domain wall problem, which threatens the standard big bang cosmology. The Lazarides-Shafi mechanism attempts to resolve this by identifying degenerate vacua through a continuous gauge symmetry. We formulate a topological quantum field theory to isolate the essential structure of the mechanism and analyze its generalized symmetry structure, including higher-form symmetries and higher-group. This framework yields a master formula for computing the domain wall number and clarifies the higher-form symmetry conditions required for complete vacuum identification in a model-independent way. Moreover, while a domain-wall-number-one scenario eliminates all higher-form global symmetries, the theory nevertheless exhibits a nontrivial four-group structure and realizes a symmetry-protected topological phase.