Representation Blocks of Conformal Fields for the $N$=4 SU(2)$_k$ Superconformal Algebras

TL;DR

This paper develops the representation theory of N=4 SU(2) superconformal algebras at arbitrary level k, introducing a basic unit of representation blocks and analyzing their transformation and screening properties.

Contribution

It introduces a new basic unit of representation blocks for N=4 SU(2) superconformal algebras and characterizes their transformation and screening operators.

Findings

Representation blocks consist of eight boson-like and eight fermion-like fields.

Transformation properties under superconformal symmetries are explicitly given.

Conditions for charge-screening operators are derived.

Abstract

The representation theories of the SU(2)$_k$-extended $N$=4 superconformal algebras (SCAs) with $arbitrary$ level $k$ are developed being based on their Feigin-Fuchs representations found recently by the present author. A basic unit of the representation blocks consisting of eight \lq\lq boson-like\rq\rq\ and eight \lq\lq fermion-like\rq\rq\ conformal fields is found to describe arbitrary representations of the $N$=4 SU(2)$_k$ SCAs, including {\it unitary} and {\it nonunitary} representations. The transformation properties of the fundamental sets of the conformal fields under the $N$=4 SU(2)$_k$ superconformal symmetries are given. Then, the whole sets of the charge-screening operators of the $N$=4 SU(2)$_k$ SCAs are identified out of the sixteen conformal fields in the basic unit of the representation blocks. The conditions for the {\it eligible} charge-screening operators are analyzed in terms of the continuous parameters which enter in our vertex-operator forms for the fundamental conformal fields of the representation blocks.