Unitary Representations of $W$ Infinity Algebras

TL;DR

This paper investigates the structure and representations of various $W$ infinity algebras, introduces a new superalgebra extension, and classifies its representations, highlighting supersymmetry properties in different series.

Contribution

It introduces a new extended superalgebra $W_{ ext{infty}}^{M,N}$ and classifies its free field realizations into continuous and discrete series with supersymmetry analysis.

Findings

Characters of representations are computed.

A new superalgebra $W_{ ext{infty}}^{M,N}$ is constructed.

Supersymmetry exists in continuous series and under specific conditions in discrete series.

Abstract

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of $W$ infinity algebras ($W_{\infty}$, $W_{1+\infty}$, $W_{\infty}^{1,1}$, $W_{\infty}^M$, $W_{1+\infty}^N$, $W_{\infty}^{M,N}$) with central charges ($2$, $1$, $3$, $2M$, $N$, $2M+N$). The characters of these representations are computed. We construct a new extended superalgebra $W_{\infty}^{M,N}$, whose bosonic sector is $W_{\infty}^M\oplus W_{1+\infty}^N$. Its representations obtained from a free field realization with central charge $2M+N$, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for $M=N$.