On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

TL;DR

This paper investigates the structure of conformal vectors in simple Z-graded vertex algebras of strong CFT type, establishing the transitive action of automorphism groups and characterizing the automorphism group of the moonshine module.

Contribution

It proves the transitivity of automorphism groups on conformal vectors and characterizes the automorphism group of the moonshine module as the Monster group.

Findings

Automorphism group acts transitively on conformal vectors in simple vertex algebras.

Uniqueness of self-dual vertex operator algebra structures for simple vertex algebras.

Automorphism group of the moonshine module is the Monster group.

Abstract

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster.