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

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.

References in corpus (2)

• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding

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• Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles
• Gauging (3+1)-dimensional topological phases: an approach from surface theories
• Decorated defect condensate, a window to unconventional quantum phase transitions in Weyl semimetals