  Symmetry protected topological (SPT) states are bulk gapped states with
gapless edge excitations protected by certain symmetries. The SPT phases in
free fermion systems, like topological insulators, can be classified by the
K-theory. However, it is not known what SPT phases exist in general interacting
systems. In this paper, we present a systematic way to construct SPT phases in
interacting bosonic systems, which allows us to identify many new SPT phases,
including three bosonic versions of topological insulators in three dimension
and one in two dimension protected by particle number conservation and time
reversal symmetry. Just as group theory allows us to construct 230 crystal
structures in 3D, we find that group cohomology theory allows us to construct
different interacting bosonic SPT phases in any dimensions and for any symmetry
groups. In particular, we are going to show how topological terms in the path
integral description of the system can be constructed from nontrivial group
cohomology classes, giving rise to exactly soluble Hamiltonians, explicit
ground state wave functions and symmetry protected gapless edge excitations.
