On gauging finite subgroups

Abstract

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