From local operators to extended objects: higher-form Symmetries and Axion quality

Abstract

The central conceptual shift underlying this thesis is the transition from viewing symmetries as acting only on local point operators to recognizing that symmetries can act on extended objects such as lines and higher-dimensional defects. This generalized perspective reorganizes the structure of gauge theories and provides a natural language for non-local constraints that are invisible in the traditional framework. We begin by reformulating ordinary (0-form) global symmetries in terms of topological symmetry defect operators (SDOs), emphasizing that their action on charged operators is understood purely by topological linking. We then develop the corresponding formalism for 1-form symmetries, where the charged objects are line operators. Using intersection theory and the topological nature of SDOs, we show why continuous 1-form symmetry groups are necessarily abelian and analyze their action on Wilson and ’t Hooft lines. In this framework, four-dimensional Maxwell theory provides a satisfying explanation about the massless 4d photon: the Coulomb phase can be interpreted as a phase with spontaneous breaking of a continuous electric 1-form symmetry, with the photon appearing as the associated Goldstone mode. With these structural tools in place, we turn to the strong CP problem and the axion. After reviewing the origin of the QCD θ-angle and the Peccei–Quinn mechanism, we focus on the axion quality problem, which revolves around the question of why the axion shift symmetry can be extraordinarily well preserved despite general expectations that quantum gravity effects violate global symmetries. The central application of this thesis is a higher-dimensional gauge theory realization of the axion in which the axion arises as a Kaluza–Klein zero mode of a higher-dimensional U(1) gauge field. In this setting, the would-be axion shift symmetry is embedded into an electric higher-form symmetry, sharply constraining which operators can contribute to an axion potential. We show how introducing electrically charged matter breaks the continuous electric 1-form symmetry to a discrete subgroup and how nonperturbative wrapped worldline configurations of heavy charged particles generate the allowed axion potential with exponential suppression. Thus, higher-form symmetry controls the structure of the axion potential while non-local dynamics controls its suppressed nature, providing exponentially good control over the axion quality problem.