Gauging Lie group symmetry in (2+1)d topological phases
Meng Cheng, Po-Shen Hsin, Chao‐Ming Jian
Abstract
We present a general algebraic framework for gauging a 0-form compact, connected Lie group symmetry in (2+1)d topological phases. Starting from a symmetry fractionalization pattern of the Lie group G <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>G</mml:mi> </mml:math> , we first extend G <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>G</mml:mi> </mml:math> to a larger symmetry group \tilde{G} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo accent="true">̃</mml:mo> </mml:mover> </mml:math> , such that there is no fractionalization with respect to \tilde{G} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo accent="true">̃</mml:mo> </mml:mover> </mml:math> in the topological phase, and the effect of gauging \tilde{G} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo accent="true">̃</mml:mo> </mml:mover> </mml:math> is to tensor the original theory with a \tilde{G} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo accent="true">̃</mml:mo> </mml:mover> </mml:math> Chern-Simons theory. To restore the desired gauge symmetry, one then has to gauge an appropriate one-form symmetry (or, condensing certain Abelian anyons) to obtain the final result. Studying the consistency of the gauging procedure leads to compatibility conditions between the symmetry fractionalization pattern and the Hall conductance. When the gauging can not be consistently done (i.e. the compatibility conditions can not be satisfied), the symmetry G <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>G</mml:mi> </mml:math> with the fractionalization pattern has an ’t Hooft anomaly and we present a general method to determine the (3+1)d topological term for the anomaly. We provide many examples, including projective simple Lie groups and unitary groups to illustrate our approach.