Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
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DOI：
10.1103/physrevd.88.045013
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发表时间：
2013-03
影响因子：
5
通讯作者：
X. Wen
中科院分区：
文献类型：
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作者：
X. Wen
本文系统地研究了玻色子和费米子弱耦合规范理论中的规范反常，规范群可以是连续的，也可以是离散的。我们显示了一个非常密切的关系规范异常和保序平凡（SPT）订单[也称为保序拓扑（SPT）订单]在一个更高的维度。利用这一思想，我们认为在d维时空中，规范反常可以用自由[H^{d+1}（G，R/Z）]\o + H_\pi^{d+1}（BG，R/Z）中的元素来描述.众所周知的阿德勒-贝尔-杰基反常是通过规范群G的群上同调类H^{d+1}（G，R/Z）的自由部分来分类的（表示为Free[H^{d+1}（G，\R/\Z）]）。我们把Adler-Bell-Jackiw规范反常以外的其他类型的规范反常称为非ABJ规范反常，其中包括维滕SU（2）整体规范反常。本文在分类空间BG中引入了\pi-上同调群H\pi^{d+1}（BG，R/Z）的概念，BG是一个Abel群，包含Tor[H^{d+1}（G，R/Z）]和拓扑上同调群H^{d+1}（BG，R/Z）作为子群.我们认为H\pi^{d+1}（BG，R/Z）分类了玻色子非ABJ规范反常，并部分分类了费米子非ABJ规范反常.使用表明规范异常与SPT相位有关的相同方法，我们也可以证明引力异常与更高维度的拓扑序（即长程纠缠模式）有关。
In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete). We show a very close relation between gauge anomalies and symmetry-protected trivial (SPT) orders [also known as symmetry-protected topological (SPT) orders] in one-higher dimensions. Using such an idea, we argue that, in d space-time dimensions, the gauge anomalies are described by the elements in Free[H^{d+1}(G,R/Z)]\oplus H_\pi^{d+1}(BG,R/Z). The well known Adler-Bell-Jackiw anomalies are classified by the free part of the group cohomology class H^{d+1}(G,R/Z) of the gauge group G (denoted as Free[H^{d+1}(G,\R/\Z)]). We refer other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as nonABJ gauge anomalies, which include Witten SU(2) global gauge anomaly. We introduce a notion of \pi-cohomology group, H_\pi^{d+1}(BG,R/Z), for the classifying space BG, which is an Abelian group and include Tor[H^{d+1}(G,R/Z)] and topological cohomology group H^{d+1}(BG,R/Z) as subgroups. We argue that H_\pi^{d+1}(BG,R/Z) classifies the bosonic nonABJ gauge anomalies, and partially classifies fermionic nonABJ anomalies. Using the same approach that shows gauge anomalies to be connected to SPT phases, we can also show that gravitational anomalies are connected to topological orders (ie patterns of long-range entanglement) in one-higher dimension.