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Quantum Duality in Electromagnetism and the Fine-Structure Constant

Clay Cordova, Kantaro Ohmori

TL;DR

The paper shows that electric-magnetic duality in Maxwell theory becomes an exact, non-invertible symmetry at rational couplings $\frac{e^{2}}{2\pi}=\frac{N_{m}}{N_{e}}$ by gauging a discrete subgroup of the one-form symmetry and composing with $\mathbb{S}$. It provides an explicit construction of topological defects $\mathcal{D}_{N_{e},N_{m}}$, analyzes their fusion rules (non-invertible in general), and studies their action on bulk and boundary operators as well as finite-volume Hilbert spaces. The authors also describe how these defects realize a wall theory with 2+1D topological sectors, discuss approximations for irrational couplings, and derive Hilbert-space equivalences and boundary-condition transformations in toroidal cavities. The results illuminate how non-invertible symmetries emerge in abelian gauge theories, with potential realizations in engineered materials and condensed-matter analogs via toric-code-like topological orders.

Abstract

We describe the interplay between electric-magnetic duality and higher symmetry in Maxwell theory. When the fine-structure constant is rational, the theory admits non-invertible symmetries which can be realized as composites of electric-magnetic duality and gauging a discrete subgroup of the one-form global symmetry. These non-invertible symmetries are approximate quantum invariances of the natural world which emerge in the infrared below the mass scale of charged particles. We construct these symmetries explicitly as topological defects and illustrate their action on local and extended operators. We also describe their action on boundary conditions and illustrate some consequences of the symmetry for Hilbert spaces of the theory defined in finite volume.

Quantum Duality in Electromagnetism and the Fine-Structure Constant

TL;DR

The paper shows that electric-magnetic duality in Maxwell theory becomes an exact, non-invertible symmetry at rational couplings by gauging a discrete subgroup of the one-form symmetry and composing with . It provides an explicit construction of topological defects , analyzes their fusion rules (non-invertible in general), and studies their action on bulk and boundary operators as well as finite-volume Hilbert spaces. The authors also describe how these defects realize a wall theory with 2+1D topological sectors, discuss approximations for irrational couplings, and derive Hilbert-space equivalences and boundary-condition transformations in toroidal cavities. The results illuminate how non-invertible symmetries emerge in abelian gauge theories, with potential realizations in engineered materials and condensed-matter analogs via toric-code-like topological orders.

Abstract

We describe the interplay between electric-magnetic duality and higher symmetry in Maxwell theory. When the fine-structure constant is rational, the theory admits non-invertible symmetries which can be realized as composites of electric-magnetic duality and gauging a discrete subgroup of the one-form global symmetry. These non-invertible symmetries are approximate quantum invariances of the natural world which emerge in the infrared below the mass scale of charged particles. We construct these symmetries explicitly as topological defects and illustrate their action on local and extended operators. We also describe their action on boundary conditions and illustrate some consequences of the symmetry for Hilbert spaces of the theory defined in finite volume.
Paper Structure (17 sections, 101 equations, 4 figures)

This paper contains 17 sections, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Electric-Magnetic Duality
  3. Interfaces and Boundary Conditions
  4. $\mathbb{S}$-Duality and Hilbert Spaces
  5. Dirichlet
  6. Neumann
  7. Dualities, Gauging, and Symmetries
  8. Self-Duality at Rational Coupling
  9. Interfaces
  10. Fusion Rules
  11. Approximating the Defect
  12. Non-Invertible Symmetry Action
  13. Action on Operators
  14. Action on Boundary Conditions
  15. Toroidal Cavity Hilbert Spaces
  16. ...and 2 more sections

Figures (4)

  • Figure 1: The $\mathbb{S}$ action on a boundary condition $\mathcal{B}$. The symmetry defect (red) collides with $\mathcal{B}$ to make a new boundary condition $\mathcal{B}\cdot \mathbb{S},$ with $A_{L}$ a new boundary field.
  • Figure 2: Illustration of \ref{['jumpformw']}. The closed cycle $\gamma_L-\gamma_R$ links with the Poincaré dual of $y$ in $W$, and the total holonomy \ref{['jumpformw']} takes a fractional value. This implies the presence of an electric one-form symmetry defect (a surface operator) on the Poincaré dual of $y$ (red).
  • Figure 3:
  • Figure 4: