Fermion-boson duality in integrable quantum field theory

TL;DR

The paper introduces a one-parameter, integrable QFT BC_n(ψ,χ,β) built from BC_n affine Toda theory coupled to massive Thirring and complex sinh-Gordon sectors, and demonstrates a duality that maps weak to strong coupling by exchanging fermions and bosons. Through perturbative analysis, factorized scattering theory, and Bethe Ansatz techniques, it derives a two-particle S-matrix for charged particles depending on a single parameter $x$, identifies neutral bound states with masses $M_a$, and shows a nontrivial $x↔1−x$ symmetry. Non-perturbative external-field calculations fix the exact relation $x(β)=β^2/(4π+β^2)$ and reveal a self-duality under $β→4π/β$, with the neutral-sector S-matrix coinciding with pure BC_n ATT and charged contributions canceling due to an underlying symmetry. This symmetry, generated by half-integer spin conserved charges forming the algebra ${rak T}_n$, enforces fermion–boson interchange at the self-dual point $β^2=4π$ and explains the exact cancellation in the neutral sector, highlighting deep connections between duality, integrability, and S-matrix structure in two-dimensional QFTs.

Abstract

We introduce and study one parameter family of integrable quantum field theories. This family has a Lagrangian description in terms of massive Thirring fermions $ψ,ψ^{\dagger}$ and charged bosons $χ,\barχ$ of complex sinh-Gordon model coupled with $BC_n$ affine Toda theory. Perturbative calculations, analysis of the factorized scattering theory and the Bethe ansatz technique are applied to show that under duality transformation, which relates weak and strong coupling regimes of the theory the fermions $ψ,ψ^{\dagger}$ transform to bosons and $χ,\barχ$ and vive versa. The scattering amplitudes of neutral particles in this theory coincide exactly with S-matrix of particles in pure $BC_n$ Toda theory, i.e. the contribution of charged bosons and fermions to these amplitudes exactly cancel each other. We describe and discuss the symmetry responsible for this compensation property.