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Exploring Non-Invertible Symmetries in Free Theories

Pierluigi Niro, Konstantinos Roumpedakis, Orr Sela

TL;DR

The paper develops a constructive framework to study non-invertible (codimension-one) topological defects in free quantum field theories, focusing on 4d Maxwell theory and the 2d free compact scalar. By requiring energy-momentum conservation across a defect, the authors derive that the most general linear mixing of $F$ and $\star F$ in Maxwell theory is an $SO(2)$ rotation with angle $\varphi$, and they construct defect Lagrangians that realize rational $\cos\varphi$ for rational bulk couplings $e^4$ and $\theta/\pi$, including a description in terms of two minimal TQFTs $\mathcal{A}^{q,p}$. They analyze how these defects act on Wilson and 't Hooft lines, showing the action is generically non-invertible even though the bulk operator map on $F$ and $\star F$ is invertible. In the 2d case, they find only four discrete maps for $d\phi$ and $\star d\phi$, with a $\mathcal{T}$-duality-like transformation realized for rational $R^2$, and the product with $\mathbb{Z}_2$, paralleling the Maxwell results. The work provides a concrete, Lagrangian-based route to non-invertible symmetries in free theories and suggests a path toward a general criterion for their existence, linking dualities, one-form symmetries, and defect fusion.

Abstract

Symmetries corresponding to local transformations of the fundamental fields that leave the action invariant give rise to (invertible) topological defects, which obey group-like fusion rules. One can construct more general (codimension-one) topological defects by specifying a map between gauge-invariant operators from one side of the defect and such operators on the other side. In this work, we apply such construction to Maxwell theory in four dimensions and to the free compact scalar theory in two dimensions. In the case of Maxwell theory, we show that a topological defect that mixes the field strength $F$ and its Hodge dual $\star F$ can be at most an $SO(2)$ rotation. For rational values of the bulk coupling and the $θ$-angle we find an explicit defect Lagrangian that realizes values of the $SO(2)$ angle $\varphi$ such that $\cos \varphi$ is also rational. We further determine the action of such defects on Wilson and 't Hooft lines and show that they are in general non-invertible. We repeat the analysis for the free compact scalar $φ$ in two dimensions. In this case we find only four discrete maps: the trivial one, a $Z_2$ map $dφ\rightarrow -dφ$, a $\mathcal{T}$-duality-like map $dφ\rightarrow i \star dφ$, and the product of the last two.

Exploring Non-Invertible Symmetries in Free Theories

TL;DR

The paper develops a constructive framework to study non-invertible (codimension-one) topological defects in free quantum field theories, focusing on 4d Maxwell theory and the 2d free compact scalar. By requiring energy-momentum conservation across a defect, the authors derive that the most general linear mixing of and in Maxwell theory is an rotation with angle , and they construct defect Lagrangians that realize rational for rational bulk couplings and , including a description in terms of two minimal TQFTs . They analyze how these defects act on Wilson and 't Hooft lines, showing the action is generically non-invertible even though the bulk operator map on and is invertible. In the 2d case, they find only four discrete maps for and , with a -duality-like transformation realized for rational , and the product with , paralleling the Maxwell results. The work provides a concrete, Lagrangian-based route to non-invertible symmetries in free theories and suggests a path toward a general criterion for their existence, linking dualities, one-form symmetries, and defect fusion.

Abstract

Symmetries corresponding to local transformations of the fundamental fields that leave the action invariant give rise to (invertible) topological defects, which obey group-like fusion rules. One can construct more general (codimension-one) topological defects by specifying a map between gauge-invariant operators from one side of the defect and such operators on the other side. In this work, we apply such construction to Maxwell theory in four dimensions and to the free compact scalar theory in two dimensions. In the case of Maxwell theory, we show that a topological defect that mixes the field strength and its Hodge dual can be at most an rotation. For rational values of the bulk coupling and the -angle we find an explicit defect Lagrangian that realizes values of the angle such that is also rational. We further determine the action of such defects on Wilson and 't Hooft lines and show that they are in general non-invertible. We repeat the analysis for the free compact scalar in two dimensions. In this case we find only four discrete maps: the trivial one, a map , a -duality-like map , and the product of the last two.
Paper Structure (15 sections, 109 equations, 3 figures)

This paper contains 15 sections, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. Defect Lagrangian
  4. Ordinary Symmetries
  5. Organization
  6. Topological defects in 4d Maxwell theory
  7. sin φ = 0
  8. sin φ not 0
  9. Action on Wilson and 't Hooft lines
  10. Topological defects in 2d scalar theory
  11. Defects from gauging and dualities
  12. Action of surface defects in Chern-Simons theory
  13. $\mathcal{S}$ defect at integer coupling
  14. Action of generic defects on lines
  15. Fusion of $\mathcal{T}$ defects

Figures (3)

  • Figure 1: A Wilson or a 't Hooft loop $L$ dragged across a non-invertible defect may become an improperly quantized disc operator. The red arrow denotes the orientation of the surface.
  • Figure 2: Piercing a defect $S$ by a line operator $L$. At the junction, there is a monopole operator $\phi$ as required by gauge invariance.
  • Figure 3: Gauge-invariant configuration of Wilson lines and twist operators on a defect $S$ in $U(1)_{2N}$ Chern-Simons theory.