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

TL;DR

This paper explores the higher-group symmetry structure of a five-dimensional topological theory using a 3-crossed module, extending concepts from four-dimensional axion electrodynamics.

Contribution

It provides a concrete verification that 3-crossed modules accurately describe the higher-group symmetry in a five-dimensional topological field theory.

Findings

Identified the 3-crossed module as the mathematical structure underlying the symmetry.

Demonstrated the consistency of gauge transformation laws with higher-group gauge theory.

Showed that background gauge invariance leads to modified gauge transformations.

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 a 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.