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

By Po-Shen Hsin, Ho Tat Lam, N. Seiberg. SciPost Physics, 2018

188 citations recorded by the source corpus.

<jats:p>We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{Z}_N</jats:tex-math><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></jats:alternatives></jats:inline-formula> one-form symmetries. A 3d topological quantum field theory (TQFT) <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{T}</jats:tex-math><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></jats:alternatives></jats:inline-formula> with such a symmetry has <jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula> special lines that generate it. The braiding of these lines and their spins are characterized by a single integer <jats:inline-formula><jats:alternatives><jats:tex-math>p</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math></jats:alternatives></jats:inline-formula> modulo <jats:inline-formula><jats:alternatives><jats:tex-math>2N</jats:tex-math><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></jats:alternatives></jats:inline-formula>. Surprisingly, if <jats:inline-formula><jats:alternatives><jats:tex-math>\gcd(N,p)=1</jats:tex-math><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></jats:alternatives></jats:inline-formula> the TQFT factorizes <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}</jats:tex-math><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></jats:alternatives></jats:inline-formula>. Here <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{T}'</jats:tex-math><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></jats:alternatives></jats:inline-formula> is a decoupled TQFT, whose lines are neutral under the global symmetry and <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{A}^{N,p}</jats:tex-math><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></jats:alternatives></jats:inline-formula> is a minimal TQFT with the <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{Z}_N</jats:tex-math><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></jats:alternatives></jats:inline-formula> one-form symmetry of label <jats:inline-formula><jats:alternatives><jats:tex-math>p</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math></jats:alternatives></jats:inline-formula>. The parameter <jats:inline-formula><jats:alternatives><jats:tex-math>p</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math></jats:alternatives></jats:inline-formula> labels the obstruction to gauging the <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{Z}_N</jats:tex-math><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></jats:alternatives></jats:inline-formula> one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When <jats:inline-formula><jats:alternatives><jats:tex-math>p=0</jats:tex-math><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></jats:alternatives></jats:inline-formula> mod <jats:inline-formula><jats:alternatives><jats:tex-math>2N</jats:tex-math><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></jats:alternatives></jats:inline-formula>, the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider <jats:inline-formula><jats:alternatives><jats:tex-math>SU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>PSU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement – probe quarks are confined. In the <jats:inline-formula><jats:alternatives><jats:tex-math>PSU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent <jats:inline-formula><jats:alternatives><jats:tex-math>\theta</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The <jats:inline-formula><jats:alternatives><jats:tex-math>PSU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory is obtained by gauging the <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{Z}_N</jats:tex-math><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></jats:alternatives></jats:inline-formula> one-form symmetry of the <jats:inline-formula><jats:alternatives><jats:tex-math>SU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the <jats:inline-formula><jats:alternatives><jats:tex-math>PSU(N)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory.</jats:p>

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/418a25f76c28bdc4e1b7dc5b1dfa487f9446358e

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Citations in the corpus

Generalized Symmetries in Condensed Matter (2022, unknown): “) Thus, gauging a 1-form symmetry in 2+1 dimensions means proliferating the worldlines of the associated anyons (16, 60); this is ‘anyon condensation’ (61).”
Anomalies in the space of coupling constants and their dynamical applications II (2019, methodology): “However, as emphasized in [37], there are other options for the theory on the interface, all with the same anomaly.”
Higher Gauging and Non-invertible Condensation Defects (2022, background): “In a bosonic QFT, the possible values of the topological spin are [89]”
Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions (2022, methodology): “The terms inside the first pair of parentheses define a 2+1d (ZN)N gauge theory living on the defect [79].”