We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.

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3+1d 中叶规范理论和对偶性的评论

我们研究了叶状规范场的性质，并在 3+1d 中构建了几种叶状场理论，这些理论描述了有物质和没有物质的叶状分形阶次，包括最近的混合分形模型。这些场论描述了与叶状规范场耦合的阿贝尔或非阿贝尔规范场理论，它们分为两类模型，我们称之为电模型和磁模型。我们表明这两类叶场理论具有二元性。我们还为子系统对称保护拓扑 (SSPT) 相构建了一个模型（使用叶状规范场和精确可解的格子哈密顿模型），这类似于单形式对称保护拓扑相，子系统对称性作用于余维- 两个子区域。我们将相应的规范 SSPT 相构建为叶状二形式规范理论。测量的 SSPT 相的一些实例是 X 立方体模型的变体，具有相同的基态简并和相同的融合，但粒子统计数据不同。