Gauging U(1) symmetry in (2+1)d topological phases

TL;DR

This paper develops a categorical framework for gauging U(1) symmetry in (2+1)d topological phases, analyzing the resulting anyon structures and effects on the phase, especially considering Hall conductance and applications to specific models.

Contribution

It introduces a novel categorical approach to U(1) gauging in modular tensor categories, extending understanding beyond finite symmetry groups and including gauge dynamics and Hall conductance effects.

Findings

Complete anyon data for gauged theories with non-zero Hall conductance

Gauging with zero Hall conductance leads to Abelian anyon condensation

Derived MTC data for $ ext{SU}(2)_k$ and $ ext{Z}_k$ parafermion models

Abstract

We study the gauging of a global U(1) symmetry in a gapped system in (2+1)d. The gauging procedure has been well-understood for a finite global symmetry group, which leads to a new gapped phase with emergent gauge structure and can be described algebraically using the mathematical framework of modular tensor category (MTC). We develop a categorical description of U(1) gauging in an MTC, taking into account the dynamics of U(1) gauge field absent in the finite group case. When the ungauged system has a non-zero Hall conductance, the gauged theory remains gapped and we determine the complete set of anyon data for the gauged theory. On the other hand, when the Hall conductance vanishes, we argue that gauging has the same effect of condensing a special Abelian anyon nucleated by inserting $2\pi$ U(1) flux. We apply our procedure to the SU(2)$_k$ MTCs and derive the full MTC data for the $\mathbb{Z}_k$ parafermion MTCs. We also discuss a dual U(1) symmetry that emerges after the original U(1) symmetry of an MTC is gauged.