Rational vertex operator algebras are finitely generated

TL;DR

This paper proves that rational vertex operator algebras are finitely generated by showing that those with algebraic Virasoro images in Zhu's algebra are finitely generated, extending to all with finite-dimensional Zhu's algebra.

Contribution

It establishes that all rational vertex operator algebras are finitely generated, a significant advancement in understanding their structure.

Findings

Rational vertex operator algebras are finitely generated.

Vertex operator algebras with finite-dimensional Zhu's algebra are finitely generated.

The algebraic nature of the Virasoro element's image implies finite generation.

Abstract

It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhu's algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhu's algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated.