Free Fields and Quasi-Finite Representation of $W_{1+\infty}$ Algebra

Abstract

We study quasi-finite representation of the $\Winf$ algebra recently proposed by Kac and Radul. When the central charge is integer, we show that they are represented by free fermions and bosonic ghosts. There are some nontrivial representations with vanishing central charge. We discuss that they may be described by large $N$ limit of topological models. We calculate their operator algebras explicitly.