The varieties of semi-conformal vectors of affine vertex operator algebras

Abstract

This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, we describe their varieties of semi-conformal vectors by some matrix equations. These matrix equations are too complicated to be solved for us. However, for affine vertex operator algebras associated to the simple Lie algebra $\mathfrak{g}$, we find the adjoint group $G$ of $\mathfrak{g}$ acts on the corresponding varieties by a natural way, which implies that such varieties should be described more clearly by studying the corresponding $G$-orbit structures. Based on above methods for general cases, as an example, considering affine vertex operator algebras associated to the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, we shall give the decompositions of $G$-orbits of varieties of their semi-conformal vectors according to different levels. Our results imply that such orbit structures depends on the levels of affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$