  We consider quantum many body systems with generalized symmetries, such as
the higher form symmetries introduced recently, and the "tensor symmetry". We
consider a general form of lattice Hamiltonians which allow a certain level of
nonlocality. Based on the assumption of dual generalized symmetries, we
explicitly construct low energy excited states. We also derive the 't Hooft
anomaly for the general Hamiltonians after "gauging" the dual generalized
symmetries. A 3d system with dual anomalous 1-form symmetries can be viewed as
the boundary of a 4d generalized symmetry protected topological (SPT) state
with 1-form symmetries. We also present a prototype example of 4d SPT state
with mixed 1-form and 0-form symmetry topological response theory as well as
its physical construction. The boundary of this SPT state can be a 3d anomalous
QED state, or an anomalous 1-form symmetry enriched topological order. Insights
are gained by dimensional compatification/reduction. After dimensional
compactification, the 3d system with N pairs of dual 1-form symmetries reduces
to a 1d system with 2N pairs of dual U(1) global symmetries, which is the
boundary of an ordinary 2d SPT state; while the 3d system with the tensor
symmetry reduces to a 1d Lifshitz theory, which is protected by the center of
mass conservation of the system.
