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The Higher Structure of Symmetries of Axion-Maxwell Theory

Michele Del Zotto, Matteo Dell'Acqua, Elias Riedel Gårding

TL;DR

The paper advances the program of extracting the higher structure of generalized symmetries in (3+1)d Axion-Maxwell theory using the worldvolume approach. It systematically classifies genuine topological defects (invertible, condensation, noninvertible), constructs noninvertible shift and electric 1-form defects from minimal 3d theories, and develops the fusion interfaces and associators that encode the full symmetry category. The key result is an explicit determination of the fusion rules, topological junctions, and F-symbols controlling the associativity of the noninvertible electric 1-form symmetry, together with a coherent framework for how these defects act on both topological and non-topological operators. This work deepens the understanding of higher-group-like and higher-categorical structures in gauge theories and provides concrete tools for analyzing dualities, anomalies, and generalized gauging in Axion-Maxwell theory.

Abstract

Generalized symmetries of quantum field theories can be characterized by topological defects/operators organized into a higher category. In this paper we consider the Axion-Maxwell field theory in four dimensions and, building on the construction of its topological defects by Choi, Lam, Shao, Hidaka, Nitta and Yokokura, we discuss field theoretical methods to compute some aspects of the higher structure of such category. In particular, we determine explicitly the generalized F-symbols for the non-invertible electric 1-form symmetry of the theory. Along the way, we clarify various aspects of the bottom-up worldvolume approach towards the calculus of defects.

The Higher Structure of Symmetries of Axion-Maxwell Theory

TL;DR

The paper advances the program of extracting the higher structure of generalized symmetries in (3+1)d Axion-Maxwell theory using the worldvolume approach. It systematically classifies genuine topological defects (invertible, condensation, noninvertible), constructs noninvertible shift and electric 1-form defects from minimal 3d theories, and develops the fusion interfaces and associators that encode the full symmetry category. The key result is an explicit determination of the fusion rules, topological junctions, and F-symbols controlling the associativity of the noninvertible electric 1-form symmetry, together with a coherent framework for how these defects act on both topological and non-topological operators. This work deepens the understanding of higher-group-like and higher-categorical structures in gauge theories and provides concrete tools for analyzing dualities, anomalies, and generalized gauging in Axion-Maxwell theory.

Abstract

Generalized symmetries of quantum field theories can be characterized by topological defects/operators organized into a higher category. In this paper we consider the Axion-Maxwell field theory in four dimensions and, building on the construction of its topological defects by Choi, Lam, Shao, Hidaka, Nitta and Yokokura, we discuss field theoretical methods to compute some aspects of the higher structure of such category. In particular, we determine explicitly the generalized F-symbols for the non-invertible electric 1-form symmetry of the theory. Along the way, we clarify various aspects of the bottom-up worldvolume approach towards the calculus of defects.

Paper Structure

This paper contains 23 sections, 3 theorems, 78 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Genuine topological defects
  4. Invertible defects
  5. Condensation defects
  6. Non-invertible defects
  7. Topological junctions
  8. Invertible symmetries
  9. Noninvertible symmetries
  10. Shift symmetry
  11. Electric symmetry
  12. (Noninvertible) higher group
  13. Non-topological operators and their junctions
  14. 't Hooft lines
  15. Axion strings
  16. ...and 8 more sections

Key Result

Theorem 1

There exists a topological interface $m_{M',M}$ between $\mathcal{D}^{(e)}_{\frac{p_1}{N_1}}\otimes\mathcal{D}^{(e)}_{\frac{p_2}{N_2}}$ and $\mathcal{D}^{(e)}_{\frac{p_1}{N_1}+\frac{p_2}{N_2}}$ given by the half-defect gauging of the decoupled, anomaly free $\mathbb{Z}_{M'}^{(0)}\subset \mathbb{Z}_{ where we denoted with $\operatorname{DW}(\mathbb{Z}_{M'})$ a $\mathbb{Z}_{M'}$ gauge theory decoupl

Figures (11)

  • Figure 1: Topological junctions between invertible and noninvertible defects.
  • Figure 2: Definition of a noninvertible electric defect as a nontrivial composition of a noninvertible interface and a topological one. The shift defects enjoy an analogous construction.
  • Figure 3: Junctions involving non-topological operators: 't Hooft lines host endpoints of Wilson lines (for compatible charges) and winding defects (\ref{['fig: Hooft junctions']}) and they link nontrivially; axion strings host endpoints of Wilson lines (for compatible charges) and magnetic defects (\ref{['fig: String junctions']}) and they link nontrivially.
  • Figure 4: The shift defect $\mathcal{D}^{(s)}_{p/n}$ acts non-inversely on the 't Hooft lines $H_m$ of the theory and attaches them to magnetic defects $\eta^{(m)}_{pm/N}$.
  • Figure 5: Consistency relation between the braiding of defects $L_i$ living on the shift symmetry operator $\mathcal{D}^{(s)}_{p/N}$ and the magnetic action on 't Hooft lines $H_m$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Remark 1
  • Example 1
  • Remark 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Remark 4
  • Example 2