AN ORBIFOLD THEORY OF GENUS ZERO ASSOCIATED TO THE SPORADIC GROUP-M24

Let V(GAMMA)l be the self-dual (or holomorphic) bosonic conformal field theory associated with the spin lattice GAMMA(l) of rank l divisible by 24. In earlier work of the authors we showed how it is possible to establish the existence and uniqueness of irreducible g-twisted sectors for V(GAMMA)l for certain automorphisms g of V(GAMMA)l, and to establish the modular invariance of the space of partition functions Z(g, h, tau) corresponding to commuting pairs g, h of elements in certain groups G of automorphisms of V(GAMMA)l. In the present work we show that if we take l = 24 and G the sporadic simple group M24, then the corresponding orbifold has the genus zero property. That is, each Z(g, h, tau) is either identically zero or a hauptmodul, i.e., it generates the field of functions on the subgroup of SL2(R) which fixes Z(g, h, tau), which then necessarily has genus zero.