Weyl groups and vertex operator algebras generated by Ising vectors satisfying $(2B,3C)$ condition

TL;DR

This paper constructs specific moonshine type vertex operator algebras generated by Ising vectors with particular subalgebra structures, linking their automorphism groups to Weyl groups of classical root systems, and analyzes their algebraic properties.

Contribution

It explicitly constructs new moonshine type vertex operator algebras generated by Ising vectors with prescribed subalgebra and automorphism group structures, connecting to Weyl groups.

Findings

Vertex operator algebras generated by Ising vectors are constructed.

The automorphism groups are isomorphic to Weyl groups of classical root systems.

The central charges and algebraic structures of these VOAs are determined.

Abstract

In this article, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors $I$ such that (1) for any $e\neq f\in I$, the subVOA $\mathrm{VOA}(e,f)$ generated by $e$ and $f$ is isomorphic to either $U_{2B}$ or $U_{3C}$; and (2)the subgroup generated by the corresponding Miyamoto involutions $\{\tau_e|\,e\in I\}$ is isomorphic to the Weyl group of a root system of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$. The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.