# Hierarchy construction and non-Abelian families of generic topological orders

## TL;DR

This paper extends hierarchy construction to generic 2+1D topological orders, introducing non-Abelian families and simplifying classification by focusing on root orders, which are non-Abelian modular extensions of Abelian group representations.

## Contribution

It generalizes hierarchy construction to non-Abelian topological orders and establishes a new equivalence relation, defining non-Abelian families and simplifying their classification.

## Key findings

- Hierarchy construction is reversible for 2+1D topological orders.
- All Abelian topological orders form the trivial non-Abelian family.
- Root topological orders are non-Abelian modular extensions of Abelian group representations.

## Abstract

We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalent class (the orbit of the hierarchy construction) as "the non-Abelian family". Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups.