On the relationship between monstrous moonshine and the uniqueness of the moonshine module

Abstract

We consider the relationship between the conjectured uniqueness of the Moonshine Module, V-h, 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 V-h 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 V-h is unique, we consider meromorphic orbifoldings of V-h and show that Monstrous Moonshine holds if and only Z(r) if the only meromorphic orbifoldings of V-h are V-h itself or the Leech theory. This constraint on the meromorphic orbifoldings of V-h therefore relates Monstrous Moonshine to the uniqueness of V-h in a new way.