Higher-form symmetries and 3-group in axion electrodynamics

TL;DR

This work identifies a rich higher-form symmetry structure in (3+1)D axion electrodynamics, showing that a semistrict 3-group (2-crossed module) governs the interplay of 0-, 1-, and 2-form symmetries. By computing correlation functions of the symmetry generators, the authors demonstrate that the Witten effect corresponds to the action of the 0-form generator on the electric 1-form sector, while the anomalous Hall effect emerges from the Peiffer lifting of two electric 1-form generators, both encoded within the 3-group. The key result is that these topological and dynamical phenomena can be understood as 3-group transformations, providing a symmetry-based, model-independent framework with potential applications to high-energy and condensed-matter contexts. The paper also outlines avenues for extending the structure to axionic domain walls, gauging higher-form symmetries, and exploring connections to string theory and quantum gravity. Overall, it establishes a principled link between topological couplings in axion electrodynamics and a higher-categorical symmetry structure with concrete physical consequences.

Abstract

We study higher-form symmetries in a low-energy effective theory of a massless axion coupled with a photon in $(3+1)$ dimensions. It is shown that the higher-form symmetries of this system are accompanied by a semistrict 3-group (2-crossed module) structure, which can be found by the correlation functions of symmetry generators of the higher-form symmetries. We argue that the Witten effect and anomalous Hall effect in the axion electrodynamics can be described in terms of 3-group transformations.

Figures (4)

• Figure 1: The correlation function in Eq. \ref{['200602.2124']}. This is the figure of a time slice of the correlation function. The black dot represents a monopole, ${\cal V}$ a time of domain wall, ${\cal S}={\cal S}_1-{\cal S}_2$ an instantaneous surface, and $\Omega_{{\cal V}-{\cal V'}}$ the space between ${\cal V}$ and ${\cal V'}$. The left hand side counts the difference of the electric flux between $T(q_{aM}, {\cal C})$ and $U_{\phi E}(e^{2\pi i n_\phi},{\cal V}_2) T(q_{aM}, {\cal C})$ by $U_{aE}(e^{2\pi i n_a},{\cal S})$. The middle side counts the magnetic flux from $T(q_{aM},{\cal C})$ by $U_{a M}(e^{2\pi i n_\phi n_a},{\cal S}\cap \Omega_{{\cal V}-{\cal V}'})$, where we used ${\cal S}_1\cap \Omega_{{\cal V}-{\cal V}'}={\cal S}\cap \Omega_{{\cal V}-{\cal V}'}$. The right panel shows the evaluation of the diagram in the middle panel. The obtained phase is $\varphi = -2\pi n_{a}n_{\phi}q_{aM} \,{\rm Link}\,({\cal S}\cap \Omega_{{\cal V}-{\cal V'}}, {\cal C})/N$.
• Figure 2: The configuration of in Eq. \ref{['200603.0238']}. This is the figure of a time slice of the correlation function. The black line represents an axionic string, ${\cal S}_1$ an instantaneous torus surface of electric flux, ${\cal S}_2={\cal S}^1_2-{\cal S}^2_2$ a time of surface operator that detects the magnetic flux, ${\cal V}_{{\cal S}_1-{\cal S}_1'}$ the space between ${\cal S}_1$ and ${\cal S}'_1$. The time slice of ${\cal S}^1_2$ is in the interior of the torus ${\cal S}_1$. The left hand side counts the difference of the magnetic flux between $V(q_{\phi M}, {\cal S}_{\rm str.})$ and $U_{a E}(e^{2\pi i n_a},{\cal S}_1) V(q_{\phi M}, {\cal S}_{\rm str.})$ by $U_{aE}(e^{2\pi i n'_a},{\cal S}_2)$. The middle side counts the winding number due to the axionic string $V(q_{\phi M}, {\cal S}_{\rm str.})$ by $U_{\phi M}(e^{-2\pi i n_a n'_a},{\cal S}_2 \cap {\cal V}_{{\cal S}_1-{\cal S}_1'})$, where we used ${\cal S}^1_2\cap {\cal V}_{{\cal S}_1-{\cal S}_1'} ={\cal S}_2\cap {\cal V}_{{\cal S}_1-{\cal S}_1'}$. The right panel shows the evaluation of the diagram in the middle panel. The obtained phase is $\varphi'= -2\pi n_an_a'q_{\phi M} \,{\rm Link}\,({\cal S}_2 \cap {\cal V}_{{\cal S}_1-{\cal S}_1'}, {\cal S}_{\rm str.})/N$.
• Figure 3: The action of $G$ on $H$. This is a figure of a time and space slice of the objects. $g$ and $g^{-1}$ are 3-dimensional objects extended to temporal and spatial directions, e.g., $\mathbb{R}\times S^2$. $h$ is an instantaneous surface object, e.g., $S^2$, and is put between $g$ and $g^{-1}$.
• Figure 4: Peiffer lifting: This is a figure of a time slice of the objects. Both of $h$ and $h'$ are surface objects. $h$ is an instantaneous surface of a torus. $h'$ is, e.g., a torus extended to temporal and spatial directions, and one circle of the time slice of the torus of $h'$ is in the interior of the torus of $h$. The other circle of the time slice of $h'$ is in the exterior of $h$, and is omitted. $\{h,h'\}$ is a line object, which is an instantaneous circle.