(3+1)-TQFTs and Topological Insulators

TL;DR

This work extends the Levin–Wen string-net construction from (2+1)D to (3+1)D by formulating a generalized string-net model based on unitary braided fusion categories (unitary premodular categories). It provides a rigorous commuting-projector Hamiltonian framework whose ground-state spaces are skein spaces on 3-manifolds and whose excitations include point-like and extended defects, analyzed via boundary and relative skein data. The paper connects these 3+1D topological orders to discrete gauge theories, BF theories, and Crane–Yetter TQFTs, and discusses holographic anomaly resolution and potential applications to 3D topological insulators and fractional topological phases, including proposed statistics like projective ribbon permutations. Overall, it offers a comprehensive blueprint for realizing and understanding 3D topological orders with both point and extended excitations, bridging lattice models, category theory, and condensed-matter physics.

Abstract

Levin-Wen models are microscopic spin models for topological phases of matter in (2+1)-dimension. We introduce a generalization of such models to (3+1)-dimension based on unitary braided fusion categories, also known as unitary premodular categories. We discuss the ground state degeneracy on 3-manifolds and statistics of excitations which include both points and defect loops. Potential connections with recently proposed fractional topological insulators and projective ribbon permutation statistics are described.