# A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons

## TL;DR

This paper proposes a classification scheme for 3+1D bosonic topological orders with only bosonic point-like excitations, linking them to finite groups and 4-cocycles, and shows they can be realized by Dijkgraaf-Witten theories.

## Contribution

It introduces a partial classification of 3+1D bosonic topological orders based on unitary pointed fusion 2-categories and connects them to Dijkgraaf-Witten gauge theories.

## Key findings

- Classifies 3+1D bosonic topological orders with bosonic point-like excitations.
- Establishes a one-to-one correspondence with finite groups and 4-cocycles.
- Demonstrates realization via Dijkgraaf-Witten gauge theories.

## Abstract

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $\omega_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.