Vertex operators and sporadic groups

TL;DR

This paper introduces enhanced vertex operator algebras and constructs examples that realize various sporadic simple groups, expanding the connection between vertex operators and finite group theory beyond the Monster group.

Contribution

It extends the concept of vertex operator algebras to include enhanced versions and constructs examples realizing multiple sporadic simple groups, including some not involved with the Monster.

Findings

Construction of enhanced vertex operator algebras

Realization of multiple sporadic simple groups

Identification of a sporadic group not involved in the Monster

Abstract

In the 1980's, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics play a role in finite group theory. In this article we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including one that is not involved in the Monster.