The varieties of Heisenberg vertex operator algebras

TL;DR

This paper introduces the concept of semi-conformal vectors and subalgebras in vertex operator algebras, analyzes their algebraic structure, and applies these ideas to characterize Heisenberg vertex operator algebras.

Contribution

It defines the set of semi-conformal vectors as an affine algebraic variety, explores their properties, and uses these to characterize Heisenberg vertex operator algebras.

Findings

The set of semi-conformal vectors forms an affine algebraic variety.

Properties of semi-conformal subalgebras are invariants of vertex operator algebras.

Characterizations of Heisenberg VOAs are obtained via properties of these varieties.

Abstract

For a vertex operator algebra $V$ with conformal vector $\omega$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,\omega)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,\omega)$, we shall understand the structure of the vertex operator algebra $(V,\omega)$. At first, we define the set $\on{Sc}(V,\omega)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,\omega)$ is an affine algebraic variety with a partial ordering and an involution map. Corresponding to each semi-conformal vector, there is a unique maximal semi-conformal vertex operator subalgebra containing it. The properties of these subalgebras are invariants of vertex operator algebras. As an example, we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras. As an application, we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.