Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

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 the following pair: a modular tensor category and a chiral central charge. In this paper, following a new line of thinking, we find that in 3+1D the classification is much simpler than it was thought to be; we propose a partial classification of topological orders for 3+1D bosonic systems: If all the pointlike excitations are bosons, then such topological orders are classified by a simpler pair (G,ω_{4}): a finite group G and its group 4-cocycle ω_{4}∈H^{4}[G;U(1)] (up to group automorphisms). Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.