Where Non-Invertible Symmetries End: Twist Defects for Electromagnetic Duality

TL;DR

This work constructs and analyzes conformal twist defects in 4d Maxwell theory that implement a non-invertible electromagnetic duality by exchanging $\vec E$ and $\vec B$ around a defect attached to a 3d topological duality defect. Using both a defect-Hilbert-space approach and defect-CFT techniques, it reveals a universal generalized free-field sector and a dynamical chiral current sector on the twist defect, with explicit defect-primaries and dimensions. The study extends to the 2d free compact boson, where a precise defect action is derived for T-duality-related non-invertible defects, twist fields are computed, and a defect ’t Hooft anomaly with a chiral $O(2)$ symmetry is discussed; rational-radius cases exhibit rich modular and fusion structures. By analyzing both Maxwell and 2d cases, the paper elucidates how non-invertible duality symmetries organize defect spectra, fusion rules, and anomaly inflows, and it connects edge-like chiral modes to the bulk duality structure, with implications for higher-form symmetries and potential applications in optics and condensed matter.

Abstract

We study novel conformal twist defects in 4d Maxwell theory, around which electric and magnetic fields are exchanged. These are codimension-2 defects living at the end of topological defects for certain non-invertible global symmetries. We determine the operator spectrum of the twist defect by solving classical electromagnetic wave equations subject to a twisted boundary condition. Using techniques from defect CFT, we show that correlation functions of these defect operators factorize into two sectors: a universal generalized free-field sector, and a chiral current sector analogous to edge modes in Chern-Simons theory. In a similar setup, we also revisit the twist fields attached to non-invertible line defects in the 2d compact boson CFT. We discuss a defect 't Hooft anomaly involving a chiral $O(2)$ symmetry, highlighting its dynamical implications.

Figures (11)

• Figure 1: Twist defect for electromagnetic duality. In 3d space, the twist defect (in red) is a string attached to a topological membrane $\cal D$ (in pink).
• Figure 2: The configuration for the conformal twist defect and the non-invertible duality defect in $\mathbb{R}^4$. The vertical red line is the 2d twist defect, which is localized at $w=\bar{w}=0$ and extends along the $(z,\bar{z})$ plane. The twist defect is attached to the 3d non-invertible topological defect $\mathcal{D}_N$ shown as the half-infinite red plane. This figure only displays the $\mathbb{R}^3$ slice defined by $z=\bar{z}$.
• Figure 3: An $S^3$ slice of the defect configuration in figure \ref{['pic_complex coordinate']}. The topological defect $\mathcal{D}_N$ is supported on the red disk, whose boundary $\rho=0$ (the red circle) is the twist defect.
• Figure 4: Poynting vector $\vec{S}$ for the electromagnetic waves in the presence of a conformal twist defect. We color those Poynting vectors with positive energy flux density in the $\psi$-direction $S_\psi>0$ in green, and those with negative $S_\psi<0$ in yellow. Given an $|s|\in \mathbb{N}+\frac{1}{4}$ or $|s|\in \mathbb{N}+\frac{3}{4}$, the energy flux density near the twist defect flows in a fixed direction along the twist defect, shown as the red circle, independent of $n$ and $\omega$. (Here we only display the Poynting vector on a $\theta=$ const slice.)
• Figure 5: DCFT two-point and three-point functions. The red plane represents the twist defect at $w=\bar{w}=0$. Left: the bulk-defect two-point function $\langle F \mathcal{O}\rangle$. Middle: the bulk-defect-defect three-point function $\langle F \mathcal{O}_2\mathcal{O}_3\rangle$ in the limit $\xi\to 0$, where the bulk point $\textbf{x}$ approaches the defect. Right: the three-point function $\langle F \mathcal{O}_2\mathcal{O}_3\rangle$ in the limit $\xi \to +\infty$, where the projection of the bulk point onto the defect plane coincides with the midpoint between $(z_2,\bar{z}_2)$ and $(z_3,\bar{z}_3)$.