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We investigate the proposition that axion-Yang-Mills systems are characterized by a 3-form gauge theory in the deep infrared regime. This hypothesis is rigorously examined by initially developing a systematic framework for analyzing 3-form gauge theory coupled to an axion, specifically focusing on its global properties. The theory consists of a BF term deformed by marginal and irrelevant operators and describes a network of vacua separated by domain walls converging at the junction of an axion string. It encompasses (-1) -, 0-, and 3-form spontaneously broken global symmetries. Utilizing this framework, in conjunction with effective field theory techniques and 't Hooft anomaly-matching conditions, we argue that the 3-form gauge theory faithfully captures the infrared physics of the axion-Yang-Mills system. The ultraviolet theory is an SU (N) Yang-Mills theory endowed with a massless Dirac fermion coupled to a complex scalar and is characterized by chiral and genuine Zₘ^ (1) 1-form center symmetries, with a mixed anomaly between them. It features two scales: the vev of the complex scalar, v, and the strong-coupling scale, , with v. Below v, the fermion decouples and two symmetries emerge: U (1) ^ (2) 2-form winding and ZN^ (1) 1-form center symmetries. As we flow below, matching the mixed anomaly necessitates introducing a dynamical 3-form gauge field of U (1) ^ (2), which appears as the incarnation of a long-range tail of the color field. The infrared theory possesses spontaneously broken chiral, emergent 3-form, and emergent (-1) -form global symmetries. It passes several checks, among which: it displays the expected restructuring in the hadronic sector upon transition between the vacua, and it is consistent under the gauging of the genuine Zₘ^ (1) ZN^ (1) symmetry.
Anber et al. (Wed,) studied this question.