Abstract
Higher-form symmetries are associated with transformations that only act on extended objects, not on point particles. Typically, higher-form symmetries live alongside ordinary, point-particle (0-form), symmetries and they can be jointly described in terms of a direct product symmetry group. However, when the actions of 0-form and higher-form symmetries become entangled, a more general mathematical structure is required, related to higher categorical groups. Systems with continuous higher-group symmetry were previously constructed in a top-down manner, descending from quantum field theories with a specific mixed ’t Hooft anomaly. I show that higher-group symmetry also naturally emerges from a bottom-up, low-energy perspective, when the physical system at hand contains at least two different given, spontaneously broken symmetries. This leads generically to a hierarchy of emergent higher-form symmetries, corresponding to the Grassmann algebra of topological currents of the theory, with an underlying higher-group structure. Examples of physical systems featuring such higher-group symmetry include superfluid mixtures and variants of axion electrodynamics.
Highlights
Higher-form symmetries live alongside ordinary, point-particle (0-form), symmetries and they can be jointly described in terms of a direct product symmetry group
A higher-group symmetry may arise from a topological coupling of otherwise disconnected sectors possessing symmetries of different degrees; this is the case for the 3-group symmetry of axion electrodynamics [37, 38], see the very recent ref. [39]
Before I show a concrete illustration of the action of the symmetry on the effective field theory (EFT) formulated in terms of the vortex variables, let me briefly mention that the above expressions for the parent action (6.1) and the symmetry transformation (6.2) can be extended to the setup of section 5 where the whole hierarchy of higher-order composite currents is taken into account, albeit with the simplifying assumption that all the Noether symmetries, and the φi variables, are 0-forms
Summary
I will focus largely on the classical aspects of continuous higher-form symmetries that will be needed later. It is hopefully clear from the discussion above that the existence of topologically conserved currents is an essential ingredient of the mechanism behind higher-group symmetries, proposed in this paper. Some further comments are deferred to the concluding section 8
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