Gapped boundary theory of 3d topological orders

Gapped boundaries of 2d nonchiral topological orders are well-understood and characterized by Lagrangian subalgebras. We extend the study to 3d and show that there exists a wide variety of options for gapped boundaries even in the simplest bosonic and fermionic toric code models. These boundaries can be organized into two classes corresponding to whether the flux string can end at the boundary. We illustrate the boundary theories from various perspectives including coupled layer construction, Walker-Wang model and field theory. Our results can be naturally generalized to other 3d topological orders.