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In this paper, we systematically study gauge anomalies in bosonic andfermionic weak-coupling gauge theories with gauge group G (which can becontinuous or discrete). We show a very close relation between gauge anomaliesand symmetry-protected trivial (SPT) orders [also known as symmetry-protectedtopological (SPT) orders] in one-higher dimensions. Using such an idea, weargue that, in d space-time dimensions, the gauge anomalies are described bythe elements in Free[H^{d+1}(G,R/Z)]\oplus H_\pi^{d+1}(BG,R/Z). The well knownAdler-Bell-Jackiw anomalies are classified by the free part of the groupcohomology class H^{d+1}(G,R/Z) of the gauge group G (denoted asFree[H^{d+1}(G,\R/\Z)]). We refer other kinds of gauge anomalies beyondAdler-Bell-Jackiw anomalies as nonABJ gauge anomalies, which include WittenSU(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 groupand include Tor[H^{d+1}(G,R/Z)] and topological cohomology groupH^{d+1}(BG,R/Z) as subgroups. We argue that H_\pi^{d+1}(BG,R/Z) classifies thebosonic nonABJ gauge anomalies, and partially classifies fermionic nonABJanomalies. Using the same approach that shows gauge anomalies to be connectedto SPT phases, we can also show that gravitational anomalies are connected totopological orders (ie patterns of long-range entanglement) in one-higherdimension.

Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders