Vertex operator algebras generated by Ising vectors of $\sigma$-type

TL;DR

This paper proves the uniqueness of certain vertex operator algebras generated by Ising vectors of $\sigma$-type and discusses conditions under which simplicity can be omitted, relating to the Griess algebra and Matsuo algebra.

Contribution

It establishes the uniqueness of OZ-type vertex operator algebras generated by Ising vectors of $\sigma$-type and explores conditions for simplicity without this assumption.

Findings

Uniqueness of OZ-type VOA generated by Ising vectors of $\sigma$-type.

Simplicity can be omitted if the Griess algebra matches the Matsuo algebra for type $A_n$.

Connections between vertex operator algebras, Griess algebra, and Matsuo algebra.

Abstract

We prove the uniqueness of the simple vertex operator algebra of OZ-type generated by Ising vectors of $\sigma$-type. We also prove that the simplicity can be omitted if the Griess algebra is isomorphic to the Matsuo algebra associated with the root system of type $A_n$.