The varieties of Heisenberg vertex operator algebras

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