Bosonization in three spatial dimensions and a 2-form gauge theory

TL;DR

This paper introduces a 3D analog of the Jordan-Wigner transformation that maps fermionic systems to 2-form gauge theories, preserving locality and depending on the spin structure, with applications to dualities and lattice models.

Contribution

It presents a novel 3D bosonization map that preserves locality and depends on spin structure, expanding duality tools in higher dimensions.

Findings

Constructed a 3D Jordan-Wigner-like transformation.

Mapped fermionic systems to 2-form $ ext{Z}_2$ gauge theories.

Provided examples of dual bosonic systems and lattice models.

Abstract

We describe a 3d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 3d spatial lattice to a 2-form $\mathbb{Z}_2$ gauge theory with an unusual Gauss law. An important property of this map is that it preserves the locality of the Hamiltonian. The map depends explicitly on the choice of a spin structure of the spatial manifold. We give examples of 3d bosonic systems dual to free fermions. We also describe the corresponding Euclidean lattice models, which is analogous to the generalized Steenrod square term in (3+1)D (compared to the Chern-Simon term in (2+1)D).