On gauging finite subgroups

<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>