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

Understand

We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$D protected by planar subsystem symmetries, which are dual to abelian fracton topological orders.

• We distinguish between weak SSPTs, which can be constructed by stacking $2+1$D 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.