Gapped boundary theories in three dimensions

TL;DR

This paper establishes a fundamental link between 3D topological field theories and boundary theories, showing that only Turaev-Viro theories admit nonzero boundary theories, with implications for gapped quantum systems.

Contribution

It proves a theorem characterizing boundary theories in 3D topological field theories and relates fusion categories to dualizability, with applications to physics.

Findings

Reshetikhin-Turaev theories admit boundary theories iff they are Turaev-Viro theories

Provides a characterization of fusion categories via dualizability

Identifies obstructions to gapped boundaries in 3D quantum systems

Abstract

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. 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 and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworked theorems and several proofs, and more.