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When $\mathbb{Z}_2$ one-form symmetry leads to non-invertible axial symmetries

Riccardo Argurio, Romain Vandepopeliere

TL;DR

This work analyzes four-dimensional non-abelian gauge theories with fermions chosen so that the surviving electric one-form symmetry is $Z_2$. It shows that a nontrivial mixed anomaly between the discrete axial symmetry and the $Z_2$ one-form symmetry yields non-invertible axial symmetry defects upon gauging, with a universal domain-wall dressing by the 3d TQFT $U(1)_2$. The results reveal a rank-dependent pattern: in many SU$(N)$ theories with two-index (anti)symmetric matter and in ${ m USp}(2N)$ ${ m N}=1$ SYM, non-invertible defects appear for certain ranks and representations, but not others, and are consistently organized by vacua structure and domain-wall fusion rules. The domain walls are decorated by $A^{2,1}$ (and related abelian TQFTs) to compensate anomalies, and the non-invertible sector emerges from the interplay between vacuum structure, line content, and gauging of the one-form symmetry. These findings illuminate how generalized symmetries, anomalies, and global form interact to constrain IR physics and reveal a universal mechanism for non-invertible symmetries in a broad class of gauge theories.

Abstract

We study non-abelian gauge theories with fermions in a representation such that the surviving electric 1-form symmetry is $\mathbb{Z}_2$. This includes $SU(N)$ gauge theories with matter in the (anti)symmetric and $N$ even, and $USp(2N)$ with a Weyl fermion in the adjoint, i.e. ${\cal N}=1$ SYM. We study the mixed 't Hooft anomaly between the discrete axial symmetry and the 1-form symmetry and show that when it is non-trivial, it leads to non-invertible symmetries upon gauging the $\mathbb{Z}_2$. The TQFT dressing the non-invertible symmetry defects is universal to all the cases we study, namely it is always a $U(1)_2$ CS theory coupled to the $\mathbb{Z}_2$ 2-form gauge field. We uncover a pattern where the presence or not of non-invertible defects depends on the rank of the gauge group.

When $\mathbb{Z}_2$ one-form symmetry leads to non-invertible axial symmetries

TL;DR

This work analyzes four-dimensional non-abelian gauge theories with fermions chosen so that the surviving electric one-form symmetry is . It shows that a nontrivial mixed anomaly between the discrete axial symmetry and the one-form symmetry yields non-invertible axial symmetry defects upon gauging, with a universal domain-wall dressing by the 3d TQFT . The results reveal a rank-dependent pattern: in many SU theories with two-index (anti)symmetric matter and in SYM, non-invertible defects appear for certain ranks and representations, but not others, and are consistently organized by vacua structure and domain-wall fusion rules. The domain walls are decorated by (and related abelian TQFTs) to compensate anomalies, and the non-invertible sector emerges from the interplay between vacuum structure, line content, and gauging of the one-form symmetry. These findings illuminate how generalized symmetries, anomalies, and global form interact to constrain IR physics and reveal a universal mechanism for non-invertible symmetries in a broad class of gauge theories.

Abstract

We study non-abelian gauge theories with fermions in a representation such that the surviving electric 1-form symmetry is . This includes gauge theories with matter in the (anti)symmetric and even, and with a Weyl fermion in the adjoint, i.e. SYM. We study the mixed 't Hooft anomaly between the discrete axial symmetry and the 1-form symmetry and show that when it is non-trivial, it leads to non-invertible symmetries upon gauging the . The TQFT dressing the non-invertible symmetry defects is universal to all the cases we study, namely it is always a CS theory coupled to the 2-form gauge field. We uncover a pattern where the presence or not of non-invertible defects depends on the rank of the gauge group.
Paper Structure (38 sections, 78 equations, 4 figures, 1 table)

This paper contains 38 sections, 78 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries for su(N) gauge algebra
  3. Line operator content of global variants
  4. SU(N)
  5. PSU(N)
  6. SU(N)/Zq
  7. Mixed axial-electric 't Hooft anomaly
  8. Non-invertible symmetries in SU(N)/Z2 with 2-index anti-symmetric representation
  9. SU(6)
  10. Before gauging
  11. After gauging : $SU(6)/ \mathbb{Z}_2$
  12. Vacuum structure
  13. Domain walls
  14. What changes for matter in the symmetric
  15. SU(8)
  16. ...and 23 more sections

Figures (4)

  • Figure 1: Vacuum structure of $SU(6)$ + AntiSym Filled points : $(0,1)$ condenses in this vacuum. Unfilled points : $(1,1)$ condenses in this vacuum. All vacua are trivially gapped.
  • Figure 2: Vacuum structure of the two gauged theories. Purple : Trivially gapped vacua. Orange : $\mathbb{Z}_2$ gauge theory.
  • Figure 3: Vacuum structure of $SU(8)$ + antiSym Filled dots : $(0,1)$ condenses in this vacuum Unfilled dots : $(1,1)$ condenses in this vacuum
  • Figure 4: Vacuum structure of the two gauged theories. Purple : Trivially gapped vacua. Orange : $\mathbb{Z}_2$ gauge theory.