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

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 ${\omega }_4$ 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 ${\omega }_4$. 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 ${\omega }_{3(b)}$, dimensionally reduced from the 4-cocycle ${\omega }_4$. 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.

36 pages, many figures, 17 tables. v3: Accepted by Phys. Rev. B. Add acknowledgements to Louis H. Kauffman

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