The elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ and its vertex operators

TL;DR

This paper constructs a free boson realization of the elliptic quantum algebra $U_{q,p}(\uhat{sl_N})$, introduces screening currents, and explicitly derives twisted vertex operators, extending known quantum affine algebra results.

Contribution

It provides the first explicit free boson realization and vertex operator constructions for the elliptic quantum algebra $U_{q,p}(sl_N)$ at any level, including new twisted expressions.

Findings

Explicit free boson realization of $U_{q,p}(sl_N)$

Construction of screening currents commuting with algebra currents

Explicit twisted vertex operators for type I and II

Abstract

We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra $U_{q}(\hat{sl_N})$. We also construct a family of screening currents, which commute with the currents of $U_{q,p}(\hat{sl_N})$ up to total q-differences. And we give explicit twisted expressions for the type $I$ and the type $II$ vertex operators of $U_{q,p}(\hat{sl_N})$ by twisting the known results of the type $I$ vertex operators of the quantum affine algebra $U_{q}(\hat{sl_N})$ and the new results of the type $II$ vertex operators of $U_{q}(\hat{sl_N})$ we obtained in this paper.