Rational Generalised Moonshine from Orbifolds

Frenkel, Lepowsky and Meurman constructed the Moonshine Module (MM) as a Z2 orbifold of the Leech Lattice Meromorphic Conformal field theory. The group of automorphisms of this theory is the 'Monster Group' M - the largest finite sporadic simple group (with order ~ 8. 1053 ). 'Monstrous Moonshine' is the famous observation that the Thompson series, corresponding to each class of M, is a hauptmodule for some genus zero fixing group. Norton considered Generalised Moonshine Functions (GMF), depending on two commuting Monster elements, and suggested that they are also hauptmodules. Using meromorphid Abelian orbifoldings of MM we identify the singularity structure of the GMF in some nontrivial cases so that the genus zero property is demonstrated and the corresponding genus zero fixing group is identified.