Uniqueness of the bosonization of the $U_q(su(2)_k)$ quantum current algebra

TL;DR

This paper investigates four different bosonizations of the $U_q(su(2)_k)$ quantum current algebra, establishing their relations and deriving a new simpler bosonization through consistency equations.

Contribution

It derives a set of consistency equations for standard form quantum currents and uncovers relations among four known bosonizations, including a new simpler one.

Findings

Recovered two known bosonizations from the equations

Derived a new, simpler bosonization

Showed all bosonizations are related via oscillator redefinitions

Abstract

Four apparently different bosonizations of the $U_q(su(2)_k)$ quantum current algebra for arbitrary level $k$ have recently been proposed in the literature. However, the relations among them have so far remained unclear except in one case. Assuming a special standard form for the $U_q(su(2)_k)$ quantum currents, we derive a set of general consistency equations that must be satisfied. As particular solutions of this set of equations, we recover two of the four bosonizations and we derive a new and simpler one. Moreover, we show that the latter three, and the remaining two bosonizations which cannot be derived directly from this set of equations since by construction they do not have the standard form, are all related to each other through some redefinitions of their Heisenberg boson oscillators.