Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations

TL;DR

This paper explores universal topological quantities in three-dimensional gapped quantum liquids, linking them to fusion algebra and non-Abelian string excitations, and generalizing modular matrices to higher dimensions.

Contribution

It introduces a framework for characterizing topological order in 3D systems using universal quantities related to fusion and statistics of excitations, extending known 2D concepts.

Findings

Universal topological quantities characterize 3D topological order.

Dimensional reduction reveals fusion and braiding of non-Abelian strings.

Generalization of modular S and T matrices to higher dimensions.

Abstract

Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group $MCG(\mathcal M)$ of the space manifold $\mathcal M$ on which the systems lives. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to fusion algebra and statistics of particle and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contains information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular $S$ and $T$ matrices to any dimension.