Global aspects of $3$-form gauge theory: implications for axion-Yang-Mills systems

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, Λ, with Λ 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 Λ, 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.