  We discuss physical constructions, and the boundary properties of various
symmetry protected topological phases that involve 1-form symmetries, from one
spatial dimension (1d) to four spatial dimensions (4d). For example, the
prototype 3d boundary state of 4d SPT states involving 1-form symmetries can be
either a gapless photon phase (quantum electrodynamics) or gapped topological
order enriched by 1-form symmetries, namely the loop excitations of these
topological orders carry nontrivial 1-form symmetry charges. This study also
serves the purpose of diagnosing anomaly of 3d states of matter. Connection
between SPT states with 1-form symmetries and condensed matter systems such as
quantum dimer models at one lower dimension will also be discussed. Whether a
quantum dimer model can have a trivial gapped phase or not depends on the
nature of its corresponding bulk state in one higher dimension.
