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The Line, the Strip and the Duality Defect

Francesco Bedogna, Salvo Mancani

TL;DR

The paper develops codim-1 condensation defects in the bulk SymTFT for the XY-plaquette and XYZ-cube, using higher gauging with discrete torsion to realize duality symmetries on the boundary. It shows that the XY-plaquette admits an exotic θ-term and a continuous non-invertible SO(2) duality at any coupling, while the XYZ-cube yields a discrete non-invertible symmetry, reflecting its absence of a continuous bulk symmetry. By analyzing both the exotic and foliated descriptions, the work demonstrates non-invertible fusion rules for the boundary duality defects and clarifies how bulk gauging renders condensation defects transparent, leaving boundary operators as genuine duality defects. The results extend the landscape of non-invertible symmetries to foliated and subsystem-symmetric systems and highlight technical challenges in defining foliated condensation defects and their measures, pointing to future work on lattice realizations and spacetime-symmetry extensions of SymTFT.

Abstract

In the Symmetry Topological Field Theories (SymTFT) that describes the exotic models XY-plaquette and XYZ-cube, we construct codim-1 condensation defects by higher gauging with discrete torsion the non-compact symmetry of the bulk. In the framework of SymTFT Mille-feuille, which captures the Lorentz-invariance breaking subsystem symmetries, these models are dual to foliated versions of Maxwell theory. We show first that the XY-plaquette model admits a $θ$-term. Then, we show these condensation defects realize non-invertible self-duality symmetries at any value of the coupling. In the XYZ-cube model such symmetry is discrete. On the other hand, we find that the XY-plaquette has a non-invertible continuous $SO(2)$ symmetry, thus extending the results in the current literature.

The Line, the Strip and the Duality Defect

TL;DR

The paper develops codim-1 condensation defects in the bulk SymTFT for the XY-plaquette and XYZ-cube, using higher gauging with discrete torsion to realize duality symmetries on the boundary. It shows that the XY-plaquette admits an exotic θ-term and a continuous non-invertible SO(2) duality at any coupling, while the XYZ-cube yields a discrete non-invertible symmetry, reflecting its absence of a continuous bulk symmetry. By analyzing both the exotic and foliated descriptions, the work demonstrates non-invertible fusion rules for the boundary duality defects and clarifies how bulk gauging renders condensation defects transparent, leaving boundary operators as genuine duality defects. The results extend the landscape of non-invertible symmetries to foliated and subsystem-symmetric systems and highlight technical challenges in defining foliated condensation defects and their measures, pointing to future work on lattice realizations and spacetime-symmetry extensions of SymTFT.

Abstract

In the Symmetry Topological Field Theories (SymTFT) that describes the exotic models XY-plaquette and XYZ-cube, we construct codim-1 condensation defects by higher gauging with discrete torsion the non-compact symmetry of the bulk. In the framework of SymTFT Mille-feuille, which captures the Lorentz-invariance breaking subsystem symmetries, these models are dual to foliated versions of Maxwell theory. We show first that the XY-plaquette model admits a -term. Then, we show these condensation defects realize non-invertible self-duality symmetries at any value of the coupling. In the XYZ-cube model such symmetry is discrete. On the other hand, we find that the XY-plaquette has a non-invertible continuous symmetry, thus extending the results in the current literature.
Paper Structure (12 sections, 87 equations, 2 figures)

This paper contains 12 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mille-feuille SymTFT and gapless exotic/foliated duality
  3. XY-plaquette model
  4. XYZ-cube model
  5. Duality of the XY plaquette model
  6. Exotic theta term
  7. Gauging and duality symmetry
  8. Condensation defects
  9. Condensation defects in 3+1d and XY plaquette
  10. Condensation defects in 4+1d and XYZ cube
  11. Conclusions and outlook
  12. Fusion of condensation defects

Figures (2)

  • Figure 1: The Mille-feuille. The vertical direction is the foliated one. Some defects of the theory will be topological only on the green layer plane.
  • Figure 2: On the condensation defect supported on $T^3$ parametrized by $(x,y,t)$, an arbitrary number $n$ of strips can be inserted at arbitrary positions of a torus $T^2 \subset T^3$, defined at fixed $t$. Some of the strips can overlap. The interval $x$ is divided into an arbitrary set of subintervals.