Free field realization of $q$-deformed primary fields for $U_q(\widehat{\sl}_2)$

TL;DR

This paper develops a $q$-deformed framework for primary fields in quantum affine algebra $U_q(\\widehat{\ ext{sl}}_2)$, introducing new operators and proving their key properties, with applications to correlation functions.

Contribution

It introduces a $q$-deformed Wakimoto module and primary fields, and proves the intertwining property of $q$-vertex operators for general spins.

Findings

Constructed a $q$-deformed primary field of spin $j$

Proved the intertwining property for $q$-vertex operators

Provided a sample correlation function calculation

Abstract

The $q$-vertex operators of Frenkel and Reshetikhin are studied by means of a $q$-deformation of the Wakimoto module for the quantum affine algebra $U_q(\widehat{\sl}_2)$ at an arbitrary level $k\ne 0,-2$. A Fock module version of the $q$-deformed primary field of spin $j$ is introduced, as well as the screening operators which (anti-)commute with the action of $U_q(\widehat{\sl}_2)$ up to a total difference of a field. A proof of the intertwining property is given for the $q$-vertex operators corresponding to the primary fields of spin $j\notin {1 \over2}\Z_{\geq0}$, which is enough to treat a general case. A sample calculation of the correlation function is also given.