Gapped Boundary Theories in Three Dimensions

We prove a theorem in 3-dimensional topological field theory: a Reshetikhin–Turaev theory admits a nonzero boundary theory iff it is a Turaev–Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. Our results rely on a (special case of) the cobordism hypothesis with singularities. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories, on 2-dualizable categories, and on internal duals may be of independent interest.