Subsystem symmetry-protected topological phases from subsystem SymTFT of 2-foliated exotic tensor gauge theory

A bstract Symmetry topological field theory (SymTFT), or topological holography, posits a correspondence between symmetries in a d -dimensional theory and topological order in a (d + 1) -dimensional theory. In this work, we extend this framework to subsystem symmetries and develop subsystem SymTFT as a systematic tool to characterize and classify subsystem symmetry-protected topological (SSPT) phases. For (2 + 1) D gapped phases, we introduce a 2-foliated (3 + 1) D exotic tensor gauge theory (which is equivalent to 2-foliated (3 + 1) D BF theory via exotic duality) as the subsystem SymTFT and systematically analyze its topological boundary conditions and linearly rigid subsystem symmetries. Taking subsystem symmetry groups G = ℤ N and G = ℤ N × ℤ M as examples, we demonstrate how to recover the classification scheme C C G = H 2 (G ×2, U (1) ) / H 2 (G, U (1) ) ) 3, which was previously derived by examining topological invariant under linear subsystem-symmetric local unitary transformations in the lattice Hamiltonian formalism. To illustrate the correspondence between field-theoretic and lattice descriptions, we further analyze ℤ 2 × ℤ 2 and ℤ N × ℤ M cluster state models as concrete examples.