Free Boson Representation of $DY_{\hbar}(sl_2)_k$ and the Deformation of the Feigin-Fuchs

TL;DR

This paper constructs a free boson realization of the Yangian double with center for sl(2) at level k, connecting it to the Feigin-Fuchs realization of affine Lie algebras in the classical limit.

Contribution

It provides a novel free boson representation of the Yangian double $DY_{h}(sl_2)_k$ and relates it to known affine Lie algebra realizations in the classical limit.

Findings

Constructed free boson realization of $DY_{h}(sl_2)_k$.

Presented screening currents commuting with the algebra.

Revealed the classical limit corresponds to Feigin-Fuchs realization.

Abstract

A realization of the Yangian double with center $DY_\hbar(sl_2)_k$ of level $k(\not=0,-2)$ in terms of free boson fields is constructed. The screening currents are also presented, which commute with $DY_\hbar(sl_2)$ modulo total difference. In the $\hbar\to 0$ limit, the currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the Feigin-Fuchs realization of affine Lie $sl(2)_k$, while the screening currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the screening currents of the affine Lie algebra $sl(2)_k$.