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

TL;DR

This paper investigates the geometric structure of varieties of semi-conformal vectors in affine vertex operator algebras, revealing how their orbit structures under group actions depend on the algebra's level, with explicit analysis for __\mathfrak{sl}_2.

Contribution

It introduces a method to analyze varieties of semi-conformal vectors via group orbit structures, providing explicit decompositions for affine vertex algebras related to simple Lie algebras.

Findings

Orbit structures depend on the level of the affine vertex algebra.

For __\mathfrak{sl}_2, the varieties are decomposed into G-orbits.

The geometric approach links algebraic properties to group actions.

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 $\mathfrak{g}$