Construction and field theory of bosonic-symmetry-protected topological states beyond group cohomology

We construct a series of bbosonic-symmetry-protected topological (BSPT) states beyond group cohomology classification using “decorated defects” approach. This construction is based on topological defects of ordinary Landau order parameters, decorated with the bosonic short-range entangled (BSRE) states in $(4k+3)d$ and $(4k+5)d$ space-time (with $k$ being nonnegative integers), which do not need any symmetry. This approach not only gives these BSPT states an intuitive physical picture, it also allows us to derive the effective field theory for all these BSPT states beyond group cohomology.