Comments on one-form global symmetries and their gauging in 3d and 4d

2018/12/11 by Po-Shen Hsin, Ho Tat Lam, Nathan Seiberg · 225 citations
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Black Holes and Theoretical Physics #Gauge theory #Geometry #Global symmetry #Homogeneous space #Mathematical physics #Mathematics #Particle physics #Physics #Quantum mechanics #Quark #Spontaneous symmetry breaking #Symmetry (geometry) #Symmetry breaking #Theoretical physics #Topological Materials and Phenomena #Topological quantum field theory #cond-mat.str-el #hep-th

paper · pdf · doi:10.21468/scipostphys.6.3.039

published in SciPost Physics 6(3) (SciPost.org) · 56 pages, 3 figures, 5 tables

arxiv created 2018/12/11 · openalex publication_date 2019/03/29 · arxiv updated 2019/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on ℤ_N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mi>N</mml:mi> </mml:msub> </mml:math> one-form symmetries. A 3d topological quantum field theory (TQFT) T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> with such a symmetry has N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> special lines that generate it. The braiding of these lines and their spins are characterized by a single integer p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> modulo 2N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . Surprisingly, if gcd(N,p)=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>gcd</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> the TQFT factorizes T=T'⊗ A^N,p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>𝒯</mml:mi> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mstyle mathvariant="script"> <mml:mi>𝒯</mml:mi> </mml:mstyle> <mml:mi>′</mml:mi> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mstyle mathvariant="script"> <mml:mi>𝒜</mml:mi> </mml:mstyle> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Here T' <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>𝒯</mml:mi> </mml:mstyle> <mml:mi>′</mml:mi> </mml:mrow> </mml:math> is a decoupled TQFT, whose lines are neutral under the global symmetry and A^N,p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mstyle mathvariant="script"> <mml:mi>𝒜</mml:mi> </mml:mstyle> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> is a minimal TQFT with the ℤ_N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mi>N</mml:mi> </mml:msub> </mml:math> one-form symmetry of label p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> . The parameter p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> labels the obstruction to gauging the ℤ_N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mi>N</mml:mi> </mml:msub> </mml:math> one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> mod

Citations

• Classification of abelian spin Chern-Simons theories
• On Domain Walls of N=1 Supersymmetric Yang-Mills in Four Dimensions
• Interferometry of non-Abelian Anyons
• On braided fusion categories I
• Generalized Global Symmetries
• Anyons in an exactly solved model and beyond
• On the Structure of Modular Categories
• Classical and quantum conformal field theory

Cited by

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• Note on generalized symmetries, gapless excitations, generalized symmetry protected topological states, and anomaly
• Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement