Peter Huston (School of Mathematics, University of Leeds) – Algebraic techniques in 3-categories of (2+1)D topological defects

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.defect networks have been proposed as a mechanism for describing fracton orders. However, determining the excitation spectra and fracton phase from the topological defect network remains a highly nontrivial task. In ongoing work with Fiona Burnell, we study a class of fracton-like defect networks in (2+1)D which give examples of non-Abelian lineons. We describe algebraic techniques for computing the excitation spectrum and ground state degeneracy and explore the interplay between the two in examples.