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

### Submission summary

 Authors (as Contributors): Meng Cheng · Po-Shen Hsin · Chao-Ming Jian
Submission information
Date submitted: 2022-12-02 02:48
Submitted by: Cheng, Meng
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Theory
• High-Energy Physics - Theory
Approach: Theoretical

### 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.

###### Current status:
Voting in preparation

### List of changes

1. We expanded the discussion in Sec. V on the connection between the gauging in the 2+1d theory and the coset construction of 1+1d CFTs.
2. We added clarifications on the relation between our description of symmetry fractionalization using group extension and the more standard description in terms of an Abelian anyon valued 2-cocycle of G.
3. We added discussions on the role of the magnetic symmetry in the gauged theory.
4. We have fixed typos throughout the text.

### Submission & Refereeing History

Resubmission 2205.15347v2 on 2 December 2022
Submission 2205.15347v1 on 7 June 2022

## Reports on this Submission

### Report

In the manuscript, the authors discuss general aspects of symmetry gauging in 2d topologically ordered phases. As far as I can judge, the paper contains new results and is interesting. It is also well written, with solid mathematical results mace accessible to a wide audience. I recommend publication.

• validity: top
• significance: top
• originality: high
• clarity: high
• formatting: excellent
• grammar: -

### Report

In this new version of the manuscript, the authors corrected the existing typos and errors and they added more discussion about the connection between the symmetry gauging in the 2+1d theory and the coset construction of 1+1d CFTs and the role of magnetic symmetry in the gauged theory. They also clarified the relation between our description of symmetry fractionalization. These results are interesting and may have applications in many areas.

I believe that the current version of the manuscript deserves to be published. I am happy to recommend it for publication.

• validity: top
• significance: high
• originality: top
• clarity: top
• formatting: excellent
• grammar: good