Gapped boundaries of (3+1)-dimensional topological order

Given a gapped boundary of a (3+1)-dimensional [(3+1)D] topological order (TO), one can stack on it a decoupled (2+1)D TO to get another boundary theory. Should one view these two boundaries as different? A natural choice would be no. Different classes of gapped boundaries of (3+1)D TO should be defined modulo the decoupled (2+1)D TOs. But is this enough? We examine the possibility of coupling the boundary of a (3+1)D TO to additional (2+1)D TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk pointlike excitations, when touching the boundary, become excitations in the added (2+1)D phase, while the stringlike excitations in the bulk may end on the boundary but with end points dressed by some other excitations in the (2+1)D phase. For a good definition of class for gapped boundaries of (3+1)D TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the stringlike excitations that can end on the boundary, whatever their end points are. A concrete example is the (3+1)D bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary, and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the m-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a (3+1)D TO char-acterized by a finite group G, different classes correspond to different normal subgroups of G. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model, and field theory.

Automated summary

This article discusses the stacking of a gapped boundary of a (3+1)D topological order (TO) with a decoupled (2+1)D TO to obtain another boundary theory. The author argues that different classes of gapped boundaries of (3+1)D TO should be defined modulo the decoupled (2+1)D TOs. Furthermore, the possibility of coupling the boundary of a (3+1)D TO to additional (2+1)D TOs or fractonic systems is examined, leading to more possibilities for gapped boundaries. The classification of gapped boundaries based on stringlike excitations is proposed.