Coulomb Gas Representations and Screening Operators of the N=4 Superconformal Algebras

TL;DR

This paper develops Coulomb gas representations for SU(2)$_k$-extended N=4 superconformal algebras, explicitly identifies all charge-screening operators, and suggests a method for proving Kac determinant formulas, advancing the algebra's BRST formalism.

Contribution

It provides the first explicit Coulomb gas representation and complete set of screening operators for the N=4 superconformal algebras with arbitrary level k.

Findings

Explicit charge-screening operators are derived.

A method for rigorous proof of Kac determinant formulas is proposed.

Results facilitate BRST formalism studies of N=4 superconformal algebras.

Abstract

The Coulomb gas representations are presented for the ${\rm SU(2)}$$_k$-extended $N$=4 superconformal algebras, incorporating the Feigin-Fuchs representation of the\break ${\rm SU(2)}$$_k$ Kac-Moody algebra with {\sl arbitrary} level $k$. Then the long-standing problem of identifying the whole set of charge-screening operators for the $N$=4 superconformal algebras is solved and their explicit expressions are given. The method of achieving a rigorous proof of the $N$=4 Kac determinant formulae following Kato and Matsuda is suggested. The complete proof for them will be given elsewhere. Our results for the screening operators also provide the basis for studying the BRST formalism of the $N$=4 superconformal algebras ${\sl {\grave a}\ la}$ Felder.