Strong 3D planar subsystem symmetry-protected topological phases and their dual fracton orders*

We classify subsystem symmetry-protected topological (SSPT) phases in 3+1D protected by planar subsystem symmetries: short-range entangled phases which are dual to long-range entangled abelian fracton topological orders via a generalized `gauging' duality. We distinguish between weak SSPTs, which can be constructed by stacking 2+1D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite abelian group. Finally, we show that fracton orders realizable via p-string condensation are dual to weak SSPTs, while those dual to strong SSPTs do not admit such a realization.