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.
Categories: cond-mat.str-el hep-th math.QA quant-ph