We consider the application of Abelian orbifold constructions in Meromorphic
Conformal Field Theory (MCFT) towards an understanding of various aspects of
Monstrous Moonshine and Generalised Moonshine. We review some of the basic
concepts in MCFT and Abelian orbifold constructions of MCFTs and summarise some
of the relevant physics lore surrounding such constructions including aspects
of the modular group, the fusion algebra and the notion of a self-dual MCFT.
The FLM Moonshine Module, V♮, is historically the first example of
such a construction being a Z2 orbifolding of the Leech lattice MCFT,
VΛ. We review the usefulness of these ideas in understanding Monstrous
Moonshine, the genus zero property for Thompson series which we have shown is
equivalent to the property that the only meromorphic Zn orbifoldings of
V♮ are VΛ and V♮ itself (assuming that
V♮ is uniquely determined by its characteristic function J(τ).
We show that these constraints on the possible Zn orbifoldings of
V♮ are also sufficient to demonstrate the genus zero property for
Generalised Moonshine functions in the simplest non-trivial prime cases by
considering Zp×Zp orbifoldings of V♮. Thus Monstrous
Moonshine implies Generalised Moonshine in these cases.Comment: Talk presented at the AMS meeting on Moonshine, the Monster and
related topics, Mt. Holyoke, June 1994, 16 pp, Plain TeX with AMS Font