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

TL;DR

This work extends a universal topological-order framework to three dimensions by analyzing simple 3D $ ext{Z}_N$ and quantum-double models on the 3-torus, revealing that ground-state overlaps furnish projective representations of $ exttt{MCG}(T^3)= ext{SL}(3, olinebreak[0] ext{Z})$ and encode information about particle and string statistics. By dimensional reduction to 2D via a small compactified direction, the authors show a precise branching of 3D data into a direct sum of 2D topological orders governed by centralizers $G_C$, enabling a unified view of 3D string fusion and braiding through 2D modular data. They provide explicit 3D $ ilde{S}$ and $ ilde{T}$ matrices, relate $ ilde{T}$ to Dehn twists and topological spin, and demonstrate how the 3D data decomposes into 2D sectors even for non-Abelian groups, with detailed treatment of the $S_3$ example. The results offer a concrete mechanism to extract 3D three-string statistics and fusion from ground-state information, paving the way for classifying 3D topological orders via their MCG representations and dimensional-reduction spectra.

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.

Figures (5)

• Figure 1: (a) Lattice site of 3D cubic lattice. $A_s$ act on spins connected to site $s$. (b) 2D plaquettes. $B_p$ acts on the four spins surrounding $p$. Choose a righthanded $(x,y,z)$ frame, and let all links be oriented wrt. to these directions. This associates a natural orientation to $2D$ plaquettes on the dual lattice.
• Figure 2: String and particle excitations. The red curve is the boundary of a membrane on the dual lattice and correspond to a string excitation. The blue links are the ones affected by the membrane operator and the green plaquettes are the ones on which $B_p$ can measure the presence of the string excitation. The green line correspond to a string operator on the lattice, in which the end point are particles. Mutual statistics between strings and particles can be calculated by creating a particle-antiparticle pair from the vacuum, moving one particle around the string excitation and annihilating the particles.
• Figure 3: The result of cutting open the 3-torus along the $x$-axis, can be represented by a hollow solid cylinder where the inner and outer surfaces are identified, but there are two boundaries along $x$. In the above, the compactified direction is $y$ and the radial direction is $z$, while the open direction is $x$. We can see the $N^3$ possible excitations on the boundaries which give rise to 3-torus ground states uppon gluing. The four first states correspond to $| \bm 1\rangle$, $| e_a\rangle$, $| m_{y,c}\rangle$ and $| m_{z,b}\rangle$.
• Figure 4: The Dehn twist $\tilde{T}$ is along the $x-y$ plane, thus it is natural to think of $T^3$ as a solid hollow 2-torus where the inner and outer boundaries are identified, here the thickened direction is $z$. In this picture, we can think of $\tilde{T}$ just as a usual Dehn twist of a 2-torus.
• Figure 5: Three string configuration, where two loops of type $b$ and $c$ are threaded by a string of type $a$.