论文标题
与HopF链接，多环和硼圈环的兼容编织，$（3+1）$ - 尺寸时空
Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
论文作者
论文摘要
拓扑激发之间的编织阶段是物理表征拓扑秩序的关键数据。在本文中，我们提供了一种现场理论方法，用于（3+1）d时空中拓扑顺序的相互兼容编织阶段的完整列表。更具体地说，考虑到离散量规组作为输入数据，本文中的拓扑激发是带有仪表电荷和带有量规通量的循环的玻感\颗粒。在这些激发中，有三类的根编织过程：粒子环编织（即，熟悉的aharonov-bohm绕过薄磁螺线管周围电荷的绕组相），多环编织[物理学。莱特牧师。 113，080403（2014）]和粒子环 - 环编织[即，在物理中编织的Borromean Rings。莱特牧师。 121，061601（2018）]。耗尽所有拓扑顺序的一种天真方法是任意结合这些根编织过程。令人惊讶的是，我们发现存在某些编织阶段无法共存的非法组合，即相互不相容。因此，由此产生的拓扑顺序是违法的，必须被排除在外。识别这些私人组合并不明显。但是，在强大的（3+1）D拓扑量子场理论（TQFTS）的帮助下，我们发现非法组合违反了规格的不变性。通过这种方式，我们能够获得所有相互兼容的编织阶段以及所有合法拓扑秩序的组合。为了说明，我们在仪表组是$ \ mathbb {z} _ {n_1}，\ mathbb {z} _ {n_1} \ times \ times \ mathbb {z} _ {n_2}，\ m athbb {z} _ {n_1} \ times \ mathbb {z} _ {n_2} \ times \ times \ mathbb {z} _ {n_3} $，和$ \ mathbb {z} _ {n_1} \ times \ times \ mathbb {z} _ {n_2} \ times \ times \ mathbb {z} _ {n_3} \ times \ times \ times \ mathbb {z} _} _ {n_4} $。最后，我们简洁地讨论了（4+1）d时空中的兼容辫子和TQFT。
Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach towards a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic \particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multi-loop braidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loop braidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601 (2018)]. A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are $\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}$, and $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}$. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.
