Quasifinite Highest Weight Modules over Super $W_{1+\infty}$ Algebra

TL;DR

This paper investigates quasifinite highest weight modules over the super $W_{1+ abla}$ algebra, characterizing their quasifiniteness via polynomials and deriving differential equations for highest weights, with applications to spectral flow and free field realization.

Contribution

It extends the analysis of quasifinite modules to the supersymmetric $W_{1+ abla}$ algebra, providing new characterizations and realizations.

Findings

Quasifiniteness characterized by polynomials.

Derived differential equations for highest weights.

Presented spectral flow and free field realization.

Abstract

We study quasifinite highest weight modules over the supersymmetric extension of the $W_{1+\infty}$ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the $(B,C)$--system, and the embedding into $\Glinf$ are also presented.