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*Summary by author Xiao-Gang Wen: *

It has been shown that bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via non-linear $\sigma$-models (NL$\sigma$Ms) of the symmetry group $G$ with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. But there are also SPT phases beyond ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$ with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. This paper shows that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification.

More precisely, the paper shows that both the pure and the mixed SPT phases can be realized by $G\times SO(\infty)$ NL$\sigma$Ms with various topological terms. Through the group cohomology ${\cal H}^d[G\times SO(\infty),\mathbb{R}/\mathbb{Z}]$, one finds that the set of constructed SPT phases in $d$-dimensional space-time are given by

$E^d(G)\rtimes \oplus_{k=1}^{d-1} {\cal

H}^k(G,\text{iTO}_L^{d-k})\oplus {\cal H}^d(G,\mathbb{R}/\mathbb{Z}) $.

Here $\text{iTO}_L^d$ is the set of the topologically-ordered phases in $d$-dimensional space-time that have no topological excitations. One finds that $\text{iTO}_L^1=\text{iTO}_L^2=\text{iTO}_L^4=\text{iTO}_L^6=0$, $\text{iTO}_L^3=\mathbb{Z}$, $\text{iTO}_L^5=\mathbb{Z}_2$, $\text{iTO}_L^7=2\mathbb{Z}$.

Black = the pure SPT phases described by ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. Blue = the mixed SPT phases described by $\oplus_{k=1}^{d-1} {\cal

H}^k(G,\text{iTO}_L^{d-k})$ but beyond ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. Red = the extra mixed SPT phases described by $E^d(G)$.

It may be possible that $G\times SO(\infty)$ NL$\sigma$Ms with various topological terms can realize all the pure and mixed SPT phases *that can be defined on arbitrary space-time manifolds*.

The NL$\sigma$M construction also gives the topological terms that fully characterize the corresponding SPT and iTO phases. Through several examples, the paper shows how can the universal physical properties of SPT phases be obtained from those topological terms.