Neveu-Schwarz and operators algebras I: Vertex operators superalgebras

TL;DR

This paper introduces a self-contained approach to vertex operator superalgebras derived from the Neveu-Schwarz algebra, constructing such structures from loop algebras of simple Lie algebras and demonstrating their applications.

Contribution

It provides an elementary, progressive method to build vertex operator superalgebras from loop algebras, linking the Neveu-Schwarz algebra to subfactors and unitary actions.

Findings

Constructed vertex operator superalgebras from loop algebras of simple Lie algebras.

Derived the Neveu-Schwarz algebra naturally within this framework.

Established a unitary action of Vir_{1/2} on discrete series.

Abstract

This paper is the first of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we present an elementary, progressive and self-contained approch to vertex operator superalgebra. We then build such a structure from the loop algebra $Lg$ of any simple finite dimensional Lie algebra $g$. The Neveu-Schwarz algebra $Vir_{1/2}$ emerges naturally on. As application, we obtain a unitary action of $Vir_{1/2}$ on the unitary discrete series of $Lg$.