Topological axion electrodynamics and 4-group symmetry

TL;DR

The paper analyzes the low-energy, gapped phase of 3+1 dimensional axion electrodynamics through a dual topological field theory with axion-photon coupling, revealing a rich hierarchy of higher-form symmetries. It identifies 0-,1-,2-, and 3-form electric symmetries and shows that their intersections host lower-dimensional topological excitations, culminating in a 4-group structure that governs all nontrivial higher-form correlations. In the bulk, a fractional Aharonov-Bohm phase signals topological order tied to the axion-photon coupling; on axionic domain walls, a distinct topological order emerges with wall-trapped anyons and a wall-specific fractional linking phase. These results provide a concrete, quantized framework for domain-wall topological order in gapped axion electrodynamics and point toward new connections between higher-form symmetries, topological order, and higher-group mathematics.

Abstract

We study higher-form symmetries and a higher group in the low energy limit of a $(3+1)$-dimensional axion electrodynamics with a massive axion and a massive photon. A topological field theory describing topological excitations with the axion-photon coupling, which we call a topological axion electrodynamics, is obtained in the low energy limit. Higher-form symmetries of the topological axion electrodynamics are specified by equations of motion and Bianchi identities. We find that there are induced anyons on the intersections of symmetry generators. By a link of worldlines of the anyons, we show that the worldvolume of an axionic domain wall is topologically ordered. We further specify the underlying mathematical structure elegantly describing all salient features of the theory to be a 4-group.

Figures (2)

• Figure 1: Intersection of temporally and spatially extended symmetry generators $U_0$ (pink sphere) and $U_1$ (orange line). This figure is a time slice of the configuration. The blue line in the right panel is a time slice of an induced static surface $U_{01}$. The blue dots denote the boundary of $\Omega_{{\cal V}_0}\cap {\cal S}_1$ on the time slice, which corresponds to an anyon on the domain wall.
• Figure 2: Intersection of symmetry generators $U_1$. This figure is a time slice of the configuration: $U_1({\cal S}_0)$ expressed by orange spheres are extended to only spatial directions, while $U_1({\cal S}_0')$ and $U_1({\cal S}_1')$ expressed by the orange lines are temporally and spatially extended. The green line in the right panel is an induced instantaneous line object $U_{11}$. The green dots denote boundaries of ${\cal V}_{{\cal S}_0} \cap {\cal S}'_1$ on the time slice.