On Gauging Symmetry of Modular Categories

TL;DR

This paper develops a mathematical framework for gauging symmetries in topological phases of matter using higher category theory, enabling the classification and construction of new topological orders.

Contribution

It formalizes the gauging process in terms of higher categories, providing explicit formulas and examples for applying this method to topological phases.

Findings

Derived a formula for the $H^4$-obstruction in gauging.

Proved properties of the gauging process.

Performed explicit gauging examples on concrete cases.

Abstract

Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group $G$, gauging is a 2-step process: first extend the UMC to a $G$-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the $H^4$-obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.