Symmetry TFT for Subsystem Symmetry

TL;DR

This paper extends symmetry topological field theory to subsystem symmetries using a 2-foliated BF theory in 3+1 dimensions, exploring dualities, boundary transformations, and non-invertible operators.

Contribution

It introduces a subsystem SymTFT framework with a 2-foliated BF theory, analyzing dualities, boundary conditions, and defect structures for subsystem symmetries.

Findings

Subsystem Kramers-Wannier and Jordan-Wigner dualities as boundary transformations.

Construction of condensation and twist defects for subsystem $SL(2,\mathbb Z_2)$.

Fusion rules for subsystem non-invertible operators derived.

Abstract

We generalize the idea of symmetry topological field theory (SymTFT) for subsystem symmetry. We propose the 2-foliated BF theory with level $N$ in $(3+1)$d as subsystem SymTFT for subsystem $\mathbb Z_N$ symmetry in $(2+1)$d. Focusing on $N=2$, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem $SL(2,\mathbb Z_2)$ symmetry of the bulk foliated BF theory. Finally, we construct the condensation defects and twist defects of $S$-transformation in the subsystem $SL(2,\mathbb Z_2)$, from which the fusion rule of subsystem non-invertible operators can be recovered.