$(3+1)$维时空中具有Hopf链、多圈和Borromean环的相容辫
Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
摘要：
拓扑激励之间的编织相位是物理表征拓扑序的关键数据。在本文中，我们提供了一个场论的方法走向一个完整的列表相互兼容的编织相的拓扑秩序在（3+1）维时空。更具体地说，考虑离散规范群作为输入数据，本文的拓扑激发是玻色子\particles携带规范电荷和环携带规范通量。在这些激励中，有三类根编织过程：颗粒环编织（即，将电荷缠绕在薄的磁性螺线管周围的熟悉的Aharonov-Bohm相）、多环编织物[Phys.Rev.Lett. 113，080403（2014）]和颗粒-环-环编织物[即，物理学评论快报中的Borromean环编织。121，061601（2018）]。穷尽所有拓扑序的一个简单方法是任意组合这些根编织过程。令人惊讶的是，我们发现存在不合理的组合，其中某些编织相不能共存，即，是互不相容的因此，由此产生的拓扑顺序是非法的，必须排除。识别这些非法组合并不明显。但是借助于强大的（3+1）维拓扑量子场论（TQFTs），我们发现不合理的组合违反了规范不变性。通过这种方式，我们能够获得所有的相互兼容的编织阶段和所有合法的拓扑顺序。为了说明这一点，我们计算了规范群为$\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}$和$\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}$时的所有细节。最后，我们简要地讨论了（4+1）维时空中的相容辫和TQFT。
英文摘要：
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.
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