3-Crossed Module Structure in the Five-Dimensional Topological Axion Electrodynamics

TL;DR

The paper analyzes a five-dimensional topological extension of axion electrodynamics and demonstrates that its symmetry content is organized by a 3-crossed module, a specific instance of a four-group gauge theory. By coupling background fields to symmetry currents via Stueckelberg terms, the authors uncover deformations of gauge transformations and modified Bianchi identities, signaling a higher-group structure. They systematically reformulate the background fields within a four-group framework and extract the 3-crossed module data, including nontrivial HL Peiffer liftings, showing consistency with gauge invariance and anomaly inflow analyses. The results validate the 3-crossed module as a concrete, physically meaningful description of higher-group symmetry in this setting and point toward even richer algebraic structures in higher dimensions.

Abstract

In this paper, we investigate the higher-group symmetry structure of a five-dimensional topological theory, which is described by a 3-crossed module. The model is obtained by an five-dimensional extension of topological axion electrodynamics in four dimensions. To study the symmetry structure, we couple background gauge fields to the symmetry currents via Stueckelberg couplings. We show that background gauge invariance requires modified gauge transformation laws, indicating the existence of a higher-group structure. Furthermore, we identify the underlying mathematical structure as a 3-crossed module by regarding the modified Stueckelberg couplings as curvatures of a higher-group gauge theory. We demonstrate that the gauge transformation laws derived from this algebraic structure are consistent with the analysis based on the gauge invariance. While our previous work introduced the concept of a 3-crossed module motivated by higher-group symmetries, this work provides concrete verification that this framework correctly captures the symmetry structure of physical theories.