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Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications

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.