A classification of 3+1D bosonic topological orders (I): the case when point-like 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 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 $ω_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.

Figures (10)

• Figure 1: (Color online) The dimension reduction of 3D space $M^2\times S^1$ to 2D space $M^2$. The top and the bottom surfaces are identified and the vertical direction is the compactified $S^1$ direction. A 3D point-like excitation (the blue dot) becomes an anyon particle in 2D. A 3D string-like excitation wrapping around $S^1$ (the red line) also becomes an anyon particle in 2D.
• Figure 2: (Color online) The untwisted sector in the dimension reduction can be realized directly on a 2D sub-manifold in 3D space without compactification.
• Figure 3: (Color online) From (a) to (b) is the braiding $c_{s,p}$ in the untwisted sector. (c)(d) are obtained from (a)(b) by shrinking strings. Shrinking thus induces a "half-braiding" isomorphism $c^\text{shr}_{s,p}$ from (c) to (d).
• Figure 4: (Color online) The braiding path of moving the string $s_2$ around $s_1$. (a), (b), (c) described the same kind of the braiding paths that can deform into each other smoothly.
• Figure 5: The fusion of boundary string-like excitations $s^\text{bdry}_{g_1} \otimes s^\text{bdry}_{g_2} = s^\text{bdry}_{g_1 g_2}$ which can be abbreviated as $g_1 \otimes g_2 = g_1 g_2$.