Exact bosonization in arbitrary dimensions

TL;DR

This paper generalizes exact bosonization to arbitrary dimensions, establishing a duality between fermionic systems and $ ext{Z}_2$ gauge theories with topological terms, preserving locality and incorporating spin structures.

Contribution

It extends the exact bosonization map to any dimension, linking fermionic systems to gauge theories with topological modifications and explicit dependence on spin structures.

Findings

Provides a duality between fermionic systems and gauge theories in arbitrary dimensions.

Derives a new formula for Stiefel-Whitney homology classes on lattices.

Shows equivalence of bosonization to adding a Steenrod square term in the path integral.

Abstract

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2d and 3d to arbitrary dimensions. This bosonization map gives a duality between any fermionic system in arbitrary $n$ spatial dimensions and a new class of $(n-1)$-form $\mathbb{Z}_2$ gauge theories in $n$ dimensions with a modified Gauss's law. This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold. A new formula for Stiefel-Whitney homology classes on lattices is derived. In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological "Steenrod square" term to the spacetime action.