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The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra $V$ with conformal vector $ω$, 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 $(V,ω)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,ω)$, we shall understand the structure of the vertex operator algebra $(V,ω)$. At first, we define the set $\on{Sc}(V,ω)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,ω)$ 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.