On gauging finite subgroups

By Yuji Tachikawa. SciPost Physics, 2017

274 citations recorded by the source corpus.

<jats:p>We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup <jats:inline-formula><jats:alternatives><jats:tex-math>A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math></jats:alternatives></jats:inline-formula> of a <jats:inline-formula><jats:alternatives><jats:tex-math>\Gamma</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Γ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-symmetric theory. Depending on how anomalous <jats:inline-formula><jats:alternatives><jats:tex-math>\Gamma</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Γ</mml:mi></mml:math></jats:alternatives></jats:inline-formula> is, we find that the symmetry of the gauged theory can be i) a direct product of <jats:inline-formula><jats:alternatives><jats:tex-math>G=\Gamma/A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mi>Γ</mml:mi><mml:mi>/</mml:mi><mml:mi>A</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> and a higher-form symmetry <jats:inline-formula><jats:alternatives><jats:tex-math>\hat A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover><mml:mi>A</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover></mml:math></jats:alternatives></jats:inline-formula> with a mixed anomaly, where <jats:inline-formula><jats:alternatives><jats:tex-math>\hat A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover><mml:mi>A</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover></mml:math></jats:alternatives></jats:inline-formula> is the Pontryagin dual of <jats:inline-formula><jats:alternatives><jats:tex-math>A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math></jats:alternatives></jats:inline-formula>; ii) an extension of the ordinary symmetry group <jats:inline-formula><jats:alternatives><jats:tex-math>G</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>G</mml:mi></mml:math></jats:alternatives></jats:inline-formula> by the higher-form symmetry <jats:inline-formula><jats:alternatives><jats:tex-math>\hat A</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover><mml:mi>A</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover></mml:math></jats:alternatives></jats:inline-formula>; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the <jats:inline-formula><jats:alternatives><jats:tex-math>H^3(G,\hat A)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mover><mml:mi>A</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.</jats:p>

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/927b285cec6ba25cc0360acc45759b38dbcc27ff

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Citations in the corpus

Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward (2022, unknown): “The two classes are related via[89, 126] α = ρ ∪ ω (C9)”
Universal Non‐Invertible Symmetries (2022, background): “The split/non-split distinction is characterized by the triviality/non-triviality of the Postnikov class which is an element in the ρ- twisted group cohomology ω ∈ H(3) ρ(Γ , Γ((1))) [14, 15].”
On 2-group global symmetries and their anomalies (2018, methodology): “A discrete analog of this construction was recently described in [14]: starting from theories with only 0-form symmetries and appropriate mixed ’t Hooft anomalies, one can construct theories with 2-group global symmetry by gauging.”
Anomalies in the space of coupling constants and their dynamical applications II (2019, background): “25 This means that the mixed anomaly between the periodicity of θ and the one-form ZN global symmetry is cancelled at the cost of making the background field B participate together with A in a higher group structure [59–61].”
Higher Gauging and Non-invertible Condensation Defects (2022, background): “2] (see also [99]), the extension is given by the mixed anomaly between the Zn subgroup and ZN/Zn before gauging.”
Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions (2017, background): “[54] had proven statements related to ours in a more mathematical set-up.”
Exploring 2-group global symmetries (2018, background): “Other recent discussions of discrete 2-group (and higher n-group) symmetries in QFT appear in [31, 16].”