Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory

Abstract

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 $G = \mathbb{Z}_N$ and $G=\mathbb{Z}_N \times \mathbb{Z}_M$ as examples, we demonstrate how to recover the classification scheme $\mathcal{C}[G] = H^{2}(G^{\times 2}, U(1)) / \left( H^2(G, U(1)) \right)^3$, 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 $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_N \times \mathbb{Z}_M$ cluster state models as concrete examples.