An orbifold theory of genus zero associated to the sporadic group \(M_{24}\)
Let \(\Gamma_{24}\) be the lattice of rank 24 whose root system is of type \(D_{24}\) and let \(V_{\Gamma_{24}}\) be the self-dual bosonic conformal field theory associated with the lattice \(\Gamma_{24}\). In this paper it is proved that if the sporadic simple group \(M_{24}\) is a group of automorphisms of \(V_{\Gamma_{24}}\) then the corresponding orbifold has the genus zero property. That is, each partition function \(Z(g,h,\tau)\) corresponding to commuting pairs \(g\), \(h\) of elements in \(M_{24}\) is either identically zero or a Hauptmodul, i.e., it generates the field of functions on the subgroup of \(\text{SL}_2(\mathbb{R})\) which fixes \(Z(g,h,\tau)\), which then necessarily has genus zero.
scientific article; zbMATH DE number 617371