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.

Automated summary

This study proves a theorem in 3-dimensional topological field theory regarding the existence of nonzero boundary theory, and characterizes fusion categories based on duality, relying on the cobordism hypothesis with singularities. The main theorem applies to 3-dimensional quantum systems in physics, providing insight into the obstruction to the existence of a boundary theory. Appendices on bordism multicategories, 2-dualizable categories, and internal duals may be of independent interest.