Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\mathrm{\infty })$ non-linear $\sigma$-models

It has been shown that the L-type 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 ${\mathcal{H}}^d(G,\mathbb{R}\mathrm{/}\mathbb{Z})$. But there are also SPT phases 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. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\times SO(\mathrm{\infty })$ NL$\sigma$Ms with various topological terms. Through the group cohomology ${\cal H}^d[G\times SO(\infty),$\mathbb{R}$/$\mathbb{Z}$]$, we find that the set of our constructed L-type SPT phases in $d$-dimensional space-time are classified by $ E^d(G)\rtimes \oplus_{k=1}^{d-1} {\cal H}^k(G,$\text{iTO}$_L^{d-k})\oplus {\cal H}^d(G,$\mathbb{R}$/$\mathbb{Z}$) $where$G$maycontaintime-reversal\mathrm{.}Here$$\text{iTO}$_L^d$isthesetoftheL-typetopologically-orderedphasesin$d$-dimensionalspace-timethathavenotopologicalexcitations,andonehas$$\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}$$. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how can the universal physical properties of SPT phases be obtained from those topological invariants.