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

We consider the relationship between the conjectured uniqueness of the Moonshine Module, V♮, 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 Zn meromorphic orbifold constructions of V♮ 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♮ is unique, we then consider meromorphic orbifoldings of V♮ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of V♮ are V♮ itself or the Leech theory. This constraint on the meromorphic orbifoldings of V♮ therefore relates Monstrous Moonshine to the uniqueness of V♮ in a new way. * EMAIL: [email protected]