Gauging Lie group symmetry in (2+1)d topological phases

TL;DR

This work develops a comprehensive two-step procedure to gauge compact, connected Lie group symmetries in 2+1d topological phases by promoting the symmetry to its universal cover $\tilde{G}$ (where symmetry fractionalization is absent) and then gauging a diagonal one-form symmetry to recover the desired $G$ gauge theory. The construction is anchored in a precise interplay between symmetry fractionalization data, the Hall response $\sigma_H$, and 't Hooft anomalies, which are diagnosed via corresponding 3+1d $G$- and $K$-SPT bulk actions. The authors provide explicit, rich examples across simple groups like ${\rm SO}(3)$, ${\rm SO}(N)$, ${\rm PSU}(N)$, ${\rm PSp}(N)$, and projective exceptional groups, highlighting when gauging is possible and when anomalies obstruct it. A parallel field-theoretic treatment using one-form symmetry clarifies the role of obstructions, offers practical recipes for constructing the gauged theories, and connects these ideas to coset constructions in 1+1d CFTs and boundary dynamics. The framework thus offers a unified, algebraic and field-theoretic toolkit for understanding symmetry-enriched topological phases with continuous symmetries and paves the way for generalizations to non-simply connected groups, higher-form symmetries, and gapless contexts.

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 $G$, we first extend $G$ to a larger symmetry group $\tilde{G}$, such that there is no fractionalization with respect to $\tilde{G}$ in the topological phase, and the effect of gauging $\tilde{G}$ is to tensor the original theory with a $\tilde{G}$ 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 $G$ 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.