On the Relationship between the Uniqueness of the Moonshine Module and Monstrous Moonshine

TL;DR

This paper explores the connection between the conjectured uniqueness of the Moonshine Module and Monstrous Moonshine, proposing that Monstrous Moonshine holds if and only if the Moonshine Module is unique and its orbifoldings are limited.

Contribution

It introduces a new relationship linking the uniqueness of the Moonshine Module to Monstrous Moonshine via orbifold constructions and centralizer correspondences.

Findings

Reproduces Thompson series for all non-Fricke classes.

Establishes a new relationship between centralizers of Monster and Conway groups.

Shows Monstrous Moonshine is equivalent to the uniqueness of the Moonshine Module.

Abstract

We consider the relationship between the conjectured uniqueness of the Moonshine Module, ${\cal V}^\natural$, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possible $Z_n$ meromorphic orbifold constructions of ${\cal V}^\natural$ based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster group $M$ together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that ${\cal V}^\natural$ is unique, we then consider meromorphic orbifoldings of ${\cal V}^\natural$ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of ${\cal V}^\natural$ give ${\cal V}^\natural$ itself or the Leech theory. This constraint on the meromorphic orbifoldings of ${\cal V}^\natural$ therefore relates Monstrous Moonshine to the uniqueness of ${\cal V}^\natural$ in a new way.