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We consider quantum many body systems with generalized symmetries, such asthe higher form symmetries introduced recently, and the "tensor symmetry". Weconsider a general form of lattice Hamiltonians which allow a certain level ofnonlocality. Based on the assumption of dual generalized symmetries, weexplicitly construct low energy excited states. We also derive the 't Hooftanomaly for the general Hamiltonians after "gauging" the dual generalizedsymmetries. A 3d system with dual anomalous 1-form symmetries can be viewed asthe boundary of a 4d generalized symmetry protected topological (SPT) statewith 1-form symmetries. We also present a prototype example of 4d SPT statewith mixed 1-form and 0-form symmetry topological response theory as well asits physical construction. The boundary of this SPT state can be a 3d anomalousQED state, or an anomalous 1-form symmetry enriched topological order. Insightsare gained by dimensional compatification/reduction. After dimensionalcompactification, the 3d system with N pairs of dual 1-form symmetries reducesto a 1d system with 2N pairs of dual U(1) global symmetries, which is theboundary of an ordinary 2d SPT state; while the 3d system with the tensorsymmetry reduces to a 1d Lifshitz theory, which is protected by the center ofmass conservation of the system.

Note on Generalized Symmetries, Gapless Excitations, Generalized Symmetry Protected Topological states, and Anomaly