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.
Highlights
Particle statistics is one of the most fundamental outcomes from quantum mechanics, which classifies particles into either boson or fermion
In terms of quantum field theories, the statistical transmutation is implemented by coupling a matter field to a Chern-Simons gauge field [3], which has been successfully employed to account for the fractional quantum Hall effect [4,5,6]
A central position is occupied by the so-called boson-fermion duality, which relates a Wilson-Fisher boson coupled to a U(1) Chern-Simons gauge field at level 1 and a free massless Dirac fermion [7,8]
Summary
Particle statistics is one of the most fundamental outcomes from quantum mechanics, which classifies particles into either boson or fermion. While the boson-fermion duality was originally proposed on the basis of the conjectured nonAbelian dualities [15], it is explicitly constructed on an array of coupled wires [16] and on a Euclidean lattice [17]. The latter approach has been extended to construct some of the non-Abelian dualities [18,19]. This work is aimed at paving the way for an unseen web of dualities in 4D by constructing an analog of the boson-fermion duality To this end, we employ the lattice construction approach developed in Ref. Because the resulting two lattice theories and their continuum limits necessarily share the same correlation functions, they constitute a novel boson-fermion duality in 4D, whose contents are to be elucidated below
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