Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices

TL;DR

This paper investigates the simplicity of a specific vertex operator algebra related to Jordan algebras, establishing conditions for simplicity based on a complex parameter and explicitly describing its maximal ideal when not simple.

Contribution

It proves the simplicity criterion for the VOA $ ext{V}_r^d$ based on the parameter $r$ and explicitly constructs the maximal ideal when the VOA is not simple.

Findings

$ ext{V}_r^d$ is simple iff $r$ is not an integer.

Explicit generator system for the maximal ideal when $r$ is an integer.

Characterization of the VOA's simplicity in relation to the parameter $r$.

Abstract

Let $r \in \BC$ be a complex number, and $d \in \BZ_{\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\Vam$ explicitly.