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

TL;DR

This paper investigates quasi-finite representations of the $W_{1+ abla}$ algebra, revealing free fermion and bosonic ghost representations at integer central charge, and exploring nontrivial zero central charge cases linked to large N topological models.

Contribution

It provides explicit constructions of quasi-finite representations of the $W_{1+ abla}$ algebra, including free fermion and ghost realizations, and analyzes their operator algebras.

Findings

Representations at integer central charge are realized by free fermions and ghosts.

Nontrivial zero central charge representations are linked to large N topological models.

Operator algebras of these representations are explicitly calculated.

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.