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 a 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 tensor symmetry reduces to a 1d Lifshitz theory, which is protected by the center of mass conservation of the system.

我们考虑具有广义对称性的量子多体系统，例如最近引入的更高形式的对称性和“张量对称性”。我们考虑允许一定程度非定域性的格子哈密顿量的一般形式。基于对偶广义对称性的假设，我们明确地构建了低能量激发态。在“测量”了对偶广义对称性之后，我们还推导出了一般哈密顿量的 't Hooft 异常。具有双重异常 1-形式对称性的 3d 系统可以被视为具有 1-形式对称性的 4d 广义对称保护拓扑 (SPT) 状态的边界。我们还展示了具有混合 1 型和 0 型对称拓扑响应理论及其物理构造的 4d SPT 状态的原型示例。这个 SPT 状态的边界可以是一个 3d 异常 QED 状态，或异常的 1 型对称性丰富的拓扑顺序。洞察力是通过维度相容/减少而获得的。尺寸紧缩后，3d 系统具有N对对偶 1 型对称性简化为具有 2 N对对偶 U(1) 全局对称性的 1d 系统，这是普通 2d SPT 状态的边界，而具有张量对称性的 3d 系统简化为 1d Lifshitz 理论，它受到系统质量守恒中心的保护。