Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory

TL;DR

<3-5 sentence high-level summary>We address how to classify and characterize string- and particle-braiding in 3+1D topological orders described by discrete gauge theories with a finite gauge group $G$ and a 4-cocycle twist $\omega_4$. The authors develop explicit SL$(3,\mathbb{Z})$ modular data, $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$, via both cocycle/path-integral and representation-theory methods, and connect these to a dimensional-reduction picture where $\mathcal{C}^{3D}_{G,\omega_4} = \bigoplus_b \mathcal{C}^{2D}_{G,\omega_{3(b)}}$. They show that the 3D braiding data decompose into 2D twisted-gauge sectors in flux-$b$ sectors and that Type II/III cocycles reproduce Abelian statistics while Type IV cocycles induce non-Abelian three-string braiding in $b\neq 0$ sectors. The framework thus enables partial classification of 3D topological orders and reveals a mechanism by which Abelian bases can become non-Abelian through cocycle twisting and multi-string braiding patterns.

Abstract

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $ω_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $ω_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $ω_{3(b)}$, dimensionally reduced from the 4-cocycle $ω_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

Figures (15)

• Figure 1: A 3D topological order $\cC^{3\tD}$ can be regarded as the direct sum of 2D topological orders $\cC^{2\tD}_b$ in different sectors $b$, as $\cC^{3\tD} = \oplus_b \cC^{2\tD}_b$, when we compactify a spatial direction $z$ into a circle. This idea is general and applicable to $\cC^{3\tD}$ without a gauge theory description. However, when $\cC^{3\tD}$ allows a gauge group $G$ description, the $b$ stands for a group element (or the conjugacy class for the non-Abelian group) of $G$. Thus $b$ acts as a gauge flux along the arrow - - -$\vartriangleright$ in the compact direction $z$. Thus, $\cC^{3\tD}$ becomes the direct sum of different $\cC^{2\tD}_b$ under distinct gauge fluxes $b$.
• Figure 2: Mutual braiding statistics following the path $1 \to 2\to 3\to 4$ along the time evolution (see Sec.\ref{['Sec:IIIc3string']}): (a) From a 2D viewpoint of dimensional reduced $\cC^{2\tD}_b$, the $2\pi$ braiding of two particles is shown. (b) The compact $z$ direction extends two particles to two closed (red, blue) strings. (c) An equivalent 3D view, the $b$ flux (along the arrow - - -$\vartriangleright$) is regarded as the monodromy caused by a third (black) string. We identify the coordinates $x,y$ and a compact $z$ to see that a full-braiding process is the one (red) string going inside to the loop of another (blue) string, and then going back from the outside. For Abelian topological orders, the mutual braiding process between two excitations (A and B) in Fig.\ref{['fig:3strings_2D_3D_xy']}(a) yields a statistical Abelian phase ${e^{\ti \theta_{\text{(A)(B)} }} \propto {\sfS}^{xy}_{\text{(A)(B)}}}$ proportional to the 2D's $\mathsf{S}^{xy}$ matrix. The dimensional-extended equivalent picture Fig.\ref{['fig:3strings_2D_3D_xy']}(c) implies that the loop-braiding yields a phase ${e^{\ti \theta_{\text{(A)(B)}, b}} \propto {\sfS}^{xy}_{b\;\text{(A)(B)}}}$ of Eq.(\ref{['eq:Sxyb']}) (up to a choice of canonical basis), where $b$ is the flux of the black string. We clarify that in both (b) and (c) our strings may carry both flux and charge. If a string carries only a pure charge, then it is effectively a point particle in 3D. If a string carries a pure flux, then it is effectively a loop of a pure string in 3D. If a string carries both charge and flux (as a dyon in 2D), then it is a loop with string fluxes attached with some charged particles in 3D. Therefore our Fig.\ref{['fig:3strings_2D_3D_xy']}(c)'s string-string braiding actually represents several braiding processes: the particle-particle, particle-loop and loop-loop braidings, all processes are threaded with a background (black) string.
• Figure 3: The illustration for ${\sfO}_{\text{(A)(B)}}= \langle \Psi_{\text{A}} | \hat{\sfO} | \Psi_{\text{B}}\rangle$. The evolution from an initial state configuration $|\Psi_{in} \rangle$ on the spatial manifold (from the top) along the time direction (the dashed line - - -) to the final state $|\Psi_{out} \rangle$ (on the bottom). For the spatial $\mathbb{T}^{d}$ torus, the mapping class group MCG$(\mathbb{T}^{d})$ is the modular SL$(d,\Z)$ transformation. We show schematically the time evolution on the spatial $\mathbb{T}^{2}$, and $\mathbb{T}^{3}$ (shown as $\mathbb{T}^{2}$ attached a $S^1$ circle on each point).
• Figure 4: The reduced 2-cocycle $\sfC^{}_{a} (b,c)$ from 3-cocycle $\omega_3$ in Eq.(\ref{['fig:Cabc']}), which triangulates a half of $\mathbb{T}^2$ and with a time interval $I$. The reduced 2-cocycle $\sfC^{}_{a} (b,c)$ from 4-cocycle $\omega_4$ in Eq.(\ref{['fig:Cabcd']}), which triangulates a half of $\mathbb{T}^3$ and with a time interval $I$. The dashed arrow $\DashedArrow$ stands for the time $t$ evolution.
• Figure 5: The vertex operator $A_v$ for the generalized twisted quantum double model in 3D. To evaluate $A_v$ operator acting on the vertex $5$, one effectively lifts $5$ to $5'$, and fill 4-cocycles $\omega$ into this geometry to compute the amplitude $\mathbf{Z}$ in Eq.(\ref{['eq:path integral']}). For this specific 3D spatial lattice surrounding vertex $5$ by $1,2,3,4$ neighbored vertices, there are four 4-cocycles $\omega$ filling in the amplitude of $A_5^{[55']}$.