Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders

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) in d space-time dimensions. We show a very close relation between gauge anomalies for gauge group G and symmetry-protected trivial (SPT) orders (also known as symmetry-protected topological (SPT) orders) with symmetry group G in one-higher dimension. The SPT phases are classified by group cohomology class H^d+1(G,R/Z). Through a more careful consideration, we argue that the gauge anomalies are described by the elements in Free[H^d+1(G,R/Z)]⊕H_π˙^d+1(BG,R/Z). The well known Adler-Bell-Jackiw anomalies are classified by the free part of H^d+1(G,R/Z) (denoted as Free[H^d+1(G,R/Z)]). We refer to other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as non-ABJ gauge anomalies, which include Witten SU(2) global gauge anomalies. We introduce a notion of π-cohomology group, H_π˙^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_π˙^d+1(BG,R/Z) classifies the bosonic non-ABJ gauge anomalies and partially classifies fermionic non-ABJ 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 (i.e., patterns of long-range entanglement) in one-higher dimension.