Topological phases of matter in (2+1)D should naturally form a 3-category, in which k-morphisms represent defects of codimension k. By the cobordism hypothesis, the 3-categories of (2+1)D topological order with a fixed anomaly are each equivalent to the 3-category of fusion categories enriched over a corresponding UMTC. In ongoing work with Fiona Burnell, we introduce algebraic techniques for concrete computations in 3-categories of (2+1)D topological order, including a tunneling approach to the classification of point defects which generalizes the use of braiding to identify anyon types. We then apply these techniques to compute ground state degeneracy and classify low energy excitations in a class of fracton-like (2+1)D topological defect networks.

We present a bicategorical state sum construction for 3-manifold invariants. Using the pivotal bicategory of spherical module categories over a spherical fusion category, we construct invariants that manifestly preserve Morita equivalence. Our main result shows that this bicategorical invariant recovers the standard Turaev–Viro invariant, thereby proving Morita invariance of Turaev–Viro invariants without appealing to the Reshetikhin–Turaev construction.

This is joint work with Jürgen Fuchs, David Jaklitsch, and Christoph Schweigert.