Generalized Moonshine and abelian orbifold constructions

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, $V^\natural$, is historically the first example of such a construction being a $Z_2$ orbifolding of the Leech lattice MCFT, $V^\Lambda$. 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 $Z_n$ orbifoldings of $V^\natural$ are $V^\Lambda$ and $V^\natural$ itself (assuming that $V^\natural$ is uniquely determined by its characteristic function $J(\tau)$. We show that these constraints on the possible $Z_n$ orbifoldings of $V^\natural$ are also sufficient to demonstrate the genus zero property for Generalised Moonshine functions in the simplest non-trivial prime cases by considering $Z_p\times Z_p$ orbifoldings of $V^\natural$. Thus Monstrous Moonshine implies Generalised Moonshine in these cases.