Vertex Operators and Matrix Elements of $U_q(su(2)_k)$ via Bosonization

TL;DR

This paper constructs bosonized vertex operators for quantum affine algebra $U_q(su(2)_k)$ at arbitrary level, providing explicit formulas, two-point functions, and demonstrating their relation to q-KZ equations and classical limits.

Contribution

It introduces a new bosonization method for vertex operators of $U_q(su(2)_k)$ applicable to any level and explicitly constructs all VOs and CVOs for $j=1/2$.

Findings

Explicit formulas for VOs and CVOs at arbitrary level

Two-point functions without screening currents for $j=1/2$

Verification that functions satisfy q-KZ equations and limits

Abstract

We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of $U_q(su(2)_k)$ for arbitrary level $k$ and representation $j\leq k/2$. Both are obtained directly as two solutions of the defining condition of vertex operators - namely that they intertwine $U_q(su(2)_k)$ modules. We construct the screening charge and present a formula for the n-point function. Specializing to $j=1/2$ we construct all VOs and CVOs explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as $k=1$; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected $k=1$ and $q=1$ limits.