On the Vertex Operators of the Elliptic Quantum Algebra $U_{q,p}(\widehat{sl_2})_{k}$}

TL;DR

This paper constructs a free boson realization of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ at any level, including vertex operators and screening currents, facilitating the calculation of correlation functions.

Contribution

It introduces a novel free boson realization of the elliptic quantum algebra and constructs explicit vertex operators and screening currents, extending the Wakimoto realization to the elliptic case.

Findings

Realization of $U_{q,p}(\widehat{sl_2})$ using free bosons.

Construction of type I and type II vertex operators.

Development of screening currents that commute with the algebra.

Abstract

A realization of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ for any given level $k$ is constructed in terms of three free boson fields and their accompanying twisted partners. It can be viewed as the elliptic deformation of Wakimoto realization. Two screening currents are constructed; they commute or anti-commute with $U_{q,p}(\widehat{sl_2})$ modulo total q-differences. The free fields realization for two types vertex operators nominated as the type $I$ and the type $II$ vertex operators are presented. The twisted version of the two types vertex operators are also obtained. They all play crucial roles in calculating correlation functions.