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

Juven Wang Massachusetts Institute of Technology Xiao-Gang Wen Perimeter Institute for Theoretical Physics

Mathematical Physics mathscidoc:1611.22002

Phys. Rev. B , 91, (035134), 2015
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 %3D's 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.
Topological quantum field theory, condensed matter, representation theory, quantum group, gauge theory, topological phases, topological order, string
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@inproceedings{juven2015non-abelian,
  title={Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory},
  author={Juven Wang, and Xiao-Gang Wen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161111081959100711612},
  booktitle={Phys. Rev. B },
  volume={91},
  number={035134},
  year={2015},
}
Juven Wang, and Xiao-Gang Wen. Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory. 2015. Vol. 91. In Phys. Rev. B . http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161111081959100711612.
https://arxiv.org/abs/1404.7854 http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.035134   -   juvenwang 2016-11-11 08:20:51 Reply
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