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

Understand

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 ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$.
• 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.