[Paper Review] Global 4-group symmetry and 't Hooft anomalies in topological axion electrodynamics

Yoshimasa Hidaka, Muneto Nitta+1 more|arXiv (Cornell University)|Aug 27, 2021

Dark Matter and Cosmic Phenomena136 references45 citations

TL;DR

This paper identifies a semi-strict 4-group (3-crossed module) structure in the low-energy effective theory of (3+1)-dimensional topological axion electrodynamics in a gapped phase, using background gauging of higher-form symmetries. It derives the associated 't Hooft anomalies and shows how these anomalies manifest in topological order, linking symmetry structures to fractional linking statistics and physical responses such as induced charges on axionic domain walls.

ABSTRACT

We study higher-form global symmetries and a higher-group structure of a low-energy limit of $(3+1)$-dimensional axion electrodynamics in a gapped phase described by a topological action. We argue that the higher-form symmetries should have a semi-strict 4-group (3-crossed module) structure by consistency conditions of couplings of the topological action to background gauge fields for the higher-form symmetries. We find possible 't Hooft anomalies for the 4-group global symmetry, and discuss physical consequences.

Motivation & Objective

• To identify the higher-form symmetry structure in the gapped phase of (3+1)-dimensional axion electrodynamics.
• To determine whether the global symmetries form a higher-group structure, specifically a 4-group (3-crossed module), via consistency of background gauge couplings.
• To compute and classify 't Hooft anomalies for the 4-group symmetry in the topological field theory.
• To explore physical consequences of the 4-group symmetry, including topological order in bulk and on axionic domain walls.
• To establish a systematic framework for higher-group gauge theories using topological field theory and background field methods.

Proposed method

• Constructs the low-energy effective action of topological axion electrodynamics in the gapped phase using a topological BF-type action with multiple gauge fields.
• Identifies 0-, 1-, 2-, and 3-form global symmetries via conserved currents derived from equations of motion and Bianchi identities.
• Performs background gauging of higher-form symmetries by coupling to (p+1)-form gauge fields, enforcing consistency of gauge transformations.
• Derives the transformation laws of symmetry generators using path-integral reparameterizations and Stokes' theorem, relating them to linking numbers.
• Analyzes the interplay between symmetry generators through nontrivial correlation functions and intersection structures.
• Uses the 3-crossed module formalism to describe the semi-strict 4-group structure, with higher-group relations encoded in gauge field transformation laws.

Experimental results

Research questions

• RQ1 What is the higher-form symmetry structure in the gapped phase of (3+1)-dimensional axion electrodynamics?
• RQ2 Does the global symmetry algebra close into a 4-group (3-crossed module) structure, and if so, what are the consistency conditions?
• RQ3 What are the 't Hooft anomalies associated with the 4-group symmetry, and how are they detected via background field coupling?
• RQ4 How do the symmetry generators and their intersections lead to topological order in the bulk and on axionic domain walls?
• RQ5 What physical observables, such as fractional linking statistics or induced charges, arise from the 4-group symmetry and anomalies?

Key findings

• The global symmetry of the gapped axion electrodynamics forms a semi-strict 4-group (3-crossed module) structure, with nontrivial higher-group relations among 0-, 1-, 2-, and 3-form symmetries.
• The theory exhibits 't Hooft anomalies for the 4-group symmetry, detected through inconsistent gauge transformations when coupling to background fields.
• The 0- and 1-form symmetry generators induce a 2-form symmetry on their intersection, with the induced generator linked to the linking number of their worldlines.
• On axionic domain walls, the 1-form and 2-form symmetry generators lead to fractional linking statistics, with a phase factor of exp(2πi q1n1 / p) for intersecting vortex lines.
• The Sikivie effect—induced electric charge on a domain wall under magnetic flux—arises as a physical consequence of the 4-group anomaly and symmetry structure.
• The 3-form symmetry generator is linked to the difference of axion field values at two points, and its transformation is tied to the linking number with a 3-cycle, confirming the 4-group consistency.