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 the varieties of semi-conformal vectors 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 g, we describe the variety of semi-conformal vectors by some matrix equations. For general cases, these matrix equations are more subtle to solve. However, for an affine vertex operator algebra associated to g, we find the adjoint group G of g acts on the corresponding variety by adjoint representation of G on g. Thus the G-orbit structures of this variety will provide information of semi-conformal vertex operator subalgebras. Based on this approach, as an example, we consider affine vertex operator algebras associated to the Lie algebra sl2(C). We shall give the decomposition of G-orbits of the variety of the semi-conformal vectors according to different levels. Our results imply that such orbit structures depend on the levels of affine vertex operator algebras.