Boson-fermion duality in four dimensions

TL;DR

This paper introduces a novel duality in four-dimensional physics linking bosonic and fermionic theories, revealing deep connections between different phases and continuum limits, with implications for understanding topological insulators and gauge theories.

Contribution

It proposes a new UV duality between lattice bosonic and fermionic theories and derives an IR duality between a free Dirac fermion and scalar QED with a vacuum angle.

Findings

Identifies a continuous phase transition corresponding to topological and trivial insulators.

Establishes a continuum IR duality between a free Dirac fermion and scalar QED with vacuum angle π.

Shows the low-energy degrees of freedom include scalar bosons and dyons forming the Dirac fermion.

Abstract

Dualities provide deep insight into physics by relating two seemingly distinct theories. Here we propose and elaborate on a novel duality between bosonic and fermionic theories in four spacetime dimensions. Starting with a Euclidean lattice action consisting of bosonic and fermionic degrees of freedom and integrating out one of them alternatively, we derive a UV duality between a Wilson fermion with self-interactions and an XY model coupled to a compact U(1) gauge field. We find a continuous phase transition between topological and trivial insulators on the fermion side corresponding to Higgs and confinement phases on the boson side. The continuum limit of each lattice theory then leads to an IR duality between a free Dirac fermion and a scalar QED with the vacuum angle $\pi$. The resulting bosonic theory proves to incorporate a scalar boson and dyons as low-energy degrees of freedom and it is their three-body composite that realizes the Dirac fermion of the fermionic theory.