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 $\mathtt{MCG}\left(\mathcal{M}\right)$ of the space manifold $\mathcal{M}$ on which the systems live. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to the fusion algebra and statistics of particles 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 contain 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.

• Received 27 September 2014
• Revised 13 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.075114