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THE UNIQUENESS OF THE MOONSHINE MODULE

In this talk we consider the relationship between the conjectured uniqueness of the Moonshine module V ♮ 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 V ♮ by considering possible alternative orbifold constructions of V ♮ from a Leech lattice compactified string. Within these constructions we find a new relationship between the centralisers of the Monster group and the Con-way group generalising an observation made by Conway and Norton. We also relate the uniqueness of V ♮ to Monstrous Moonshine and argue that given this uniqueness, then the genus zero properties hold if and only if orbifolding V ♮ with respect to a monster element reproduces V ♮ or the Leech theory.