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 FreeH^d+1 (G, R/Z) { {H_}}^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 FreeH^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 -cohomology group, { {H_}}^d+1 (BG, R/Z), for the classifying space BG, which is an Abelian group and include TorH^d+1 (G, R/Z) and topological cohomology group H^d+1 (BG, R/Z) as subgroups. We argue that { {H_}}^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.