A Quantum Analogue of the Boson-Fermion Correspondence

TL;DR

This paper explores the classical and quantum boson-fermion correspondence within the $ff$ current algebra at level 2, extending the classical relation to the quantum algebra through new fermionic realizations.

Contribution

It introduces a fermionic realization of the quantum current algebra $U_q(ff)$ at level 2 and extends the classical boson-fermion correspondence to the quantum setting.

Findings

Classical boson-fermion correspondence is reviewed for $ff$ at level 2.

A fermionic realization of the quantum algebra $U_q(ff)$ at level 2 is derived.

The classical correspondence is extended to the quantum case by comparing realizations.

Abstract

We review the classical boson-fermion correspondence in the context of the $\widehat{sl(2)}$ current algebra at level 2. This particular algebra is ideal to exhibit this correspondence because it can be realized either in terms of three real bosonic fields or in terms of one real and one complex fermionic fields. We also derive a fermionic realization of the quantum current algebra $U_q(\widehat{sl(2)})$ at level 2 and by comparing this realization with the existing bosonic one we extend the classical correspondence to the quantum case.