Free Field Representations and Screening Operators for the $N=4$ Doubly Extended Superconformal Algebras

TL;DR

This paper constructs explicit free field representations and screening operators for the $N=4$ doubly extended superconformal algebra, revealing its structure and generalizing previous results in superconformal algebra theory.

Contribution

It provides the first explicit free field realization and screening operators for the $N=4$ doubly extended superconformal algebra, extending the understanding of its structure and representations.

Findings

Explicit free field representations for $ ilde{ m{A}}_{ m{ extgamma}}$

Identification of screening operators and singular vectors

Conjecture for the Kac determinant in this algebra

Abstract

We present explicit free field representations for the $N=4$ doubly extended superconformal algebra, $\tilde{\cal{A}}_{\gamma}$. This algebra generalizes and contains all previous $N=4$ superconformal algebras. We have found $\tilde{\cal{A}}_{\gamma}$ to be obtained by hamiltonian reduction of the Lie superalgebra $D(2|1;\alpha)$. In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to present a natural conjecture for the Kac determinant generalizing a previous conjecture by Kent and Riggs for the singly extended case. The results support and illuminate several aspects of the characters of this algebra previously obtained by Taormina and one of us.