Monstrous Moonshine and the uniqueness of the Moonshine module

TL;DR

This paper explores the conjectured uniqueness of the Moonshine module and its connection to Monstrous Moonshine, proposing that orbifold constructions from the Leech lattice support the module's uniqueness and the genus zero property.

Contribution

It provides evidence for the uniqueness of the Moonshine module through orbifold constructions and establishes new relationships between centralisers of the Monster and Conway groups.

Findings

Orbifold constructions support the Moonshine module's uniqueness.

New relationships between Monster and Conway group centralisers.

Genus zero property linked to orbifolding the Moonshine module.

Abstract

In this talk we consider the relationship between the conjectured uniqueness of the Moonshine module of Frenkel, Lepowsky and Meurman and Monstrous Moonshine, the genus zero property for Thompson series discovered by Conway and Norton. We discuss some evidence to support the uniqueness of the Moonshine module by considering possible alternative orbifold constructions from a Leech lattice compactified string. Within these constructions we find a new relationship between the centralisers of the Monster group and the Conway group generalising an observation made by Conway and Norton. We also relate the uniqueness of the Moonshine module to Monstrous Moonshine and argue that given this uniqueness, then the genus zero properties hold if and only if orbifolding the Moonshine module with respect to a Monster element reproduces the Moonshine module or the Leech theory. (Talk presented at the Nato Advanced Research Workshop on `Low dimensional topology and quantum field theory`, Cambridge, 6-13 Sept 1992)