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

Abstract

We consider the relationship between the conjectured uniqueness of the Moonshine Module, ${\mathcal{V}}^{♮}$, 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 $Z_n$ meromorphic orbifold constructions of ${\mathcal{V}}^{♮}$ 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 ${\mathcal{V}}^{♮}$ is unique, we then consider meromorphic orbifoldings of ${\mathcal{V}}^{♮}$ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of ${\mathcal{V}}^{♮}$ give ${\mathcal{V}}^{♮}$ itself or the Leech theory. This constraint on the meromorphic orbifoldings of ${\mathcal{V}}^{♮}$ therefore relates Monstrous Moonshine to the uniqueness of ${\mathcal{V}}^{♮}$ in a new way.

53 pages, PlainTex, DIAS-STP-93-09

References in corpus (1)

• Discussion of Self-Dual c=24 Conformal Field Theories

Cited by in corpus (19)

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• Monstrous branes
• Universal near-horizon conformal structure and black hole entropy
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• Mathieu Moonshine and Orbifold K3s
• Duality Defect of the Monster CFT
• Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module
• Fricke Lie algebras and the genus zero property in Moonshine
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• Character relations and replication identities in 2d Conformal Field Theory
• Monstrous Moonshine over Z?
• Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions
• 2A-orbifold construction and the baby-monster vertex operator superalgebra