Boson-fermion duality with subsystem symmetry

TL;DR

This paper establishes an exact duality in (2+1)d between bosonic and fermionic theories with subsystem symmetries, generalizing known dualities and introducing a subsystem Arf invariant with a foliation structure.

Contribution

It introduces a novel duality framework for (2+1)d systems with subsystem symmetries and generalizes the Jordan-Wigner map to this context, including the subsystem Arf invariant.

Findings

Established lattice duality using generalized Jordan-Wigner map

Mapped twist and symmetry sectors explicitly

Introduced subsystem Arf invariant with foliation structure

Abstract

We explore an exact duality in $(2+1)$d between the fermionization of a bosonic theory with a $\mathbb{Z}_2$ subsystem symmetry and a fermionic theory with a $\mathbb{Z}_2$ subsystem fermion parity symmetry. A typical example is the duality between the fermionization of the plaquette Ising model and the plaquette fermion model. We first revisit the standard boson-fermion duality in $(1+1)$d with a $\mathbb{Z}_2$ 0-from symmetry, presenting in a way generalizable to $(2+1)$d. We proceed to $(2+1)$d with a $\mathbb{Z}_2$ subsystem symmetry and establish the exact duality on the lattice by using the generalized Jordan-Wigner map, with a careful discussion on the mapping of the twist and symmetry sectors. This motivates us to introduce the subsystem Arf invariant, which exhibits a foliation structure.