Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
By X. Wen. 2013
196 citations recorded by the source corpus.
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.
Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/7fea3133f37dd6b2bee02e5e44be609535028444
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Citations in the corpus
- Symmetry protected topological phases and generalized cohomology (2017, background): “The general classification of Symmetry Protected Topological phases of matter is an important problem in theoretical physics and mathematics [Kit01, Kit09, GW09, PBTO12, CGW10, CGLW13, FK11, CLW11, CGW11, VS13, GW14, Wen13, WS14, Wen14, HW14, CFV14, BCFV14, Kit13, Kap14, KTTW15, FH16, Xio17].”
- Topological Classification of Crystalline Insulators through Band Structure Combinatorics (2016, unknown): “While the theoretical description of symmetry-protected topological phases (SPTs) is connected to exotic field theories and branches of mathematics [4], their interplay between topology and symmetry also has direct and prominent experimental consequences.”