An orbifold theory of genus zero associated to the sporadic groupM ₂₄

Abstract

Let$V_{\Gamma _l }$ be the self-dual (or holomorphic) bosonic conformal field theory associated with the spin lattice Γ _l of rankl divisible by 24. In earlier work of the authors we showed how it is possible to establish the existence and uniqueness of irreducibleg-twisted sectors for$V_{\Gamma _l }$, for certain automorphismsg of$V_{\Gamma _l }$, and to establish the modular invariance of the space of partition functionsZ(g, h, τ) corresponding to commuting pairsg, h of elements in certain groupsG of automorphisms of$V_{\Gamma _l }$. In the present work we show that if we takel=24 andG the sporadic simple groupM ₂₄, then the corresponding orbifold has thegenus zero property. That is, eachZ(g, h, τ) is either identically zero or ahauptmodul, i.e., it generates the field of functions on the subgroup ofSL ₂(ℝ) which fixesZ(g, h, τ), which then necessarily has genus zero.