Vertex operators, and three sporadic groups
We introduce the notion of super vertex operator algebra with enhanced conformal structure, which is a refinement of the notion of super vertex operator algebra, and we present applications of this notion to three sporadic simple groups: the largest sporadic group of Conway, the sporadic group of Suzuki, and the sporadic group of Rudvalis. For the Conway group we construct what may be considered a natural super-analogue of the Moonshine Module, where the role of the Virasoro algebra is now played by the N = 1 Virasoro superalgebra, and we find that the full group of automorphisms is Conway's largest sporadic group. We also verify a uniqueness result for this object, which is directly analogous to that conjectured to hold for the Moonshine Module. Replacing the N = 1 Virasoro superalgebra with yet richer extensions of the Virasoro algebra, one may obtain realizations of other sporadic simple groups, and we provide full details for the case of the Suzuki group, obtaining a super vertex operator algebra with enhanced conformal structure whose full automorphism group is a triple cover of this group. We then show that vertex operators have application to sporadic groups beyond the Monster by constructing a super vertex operator algebra with enhanced conformal structure whose full automorphism group is a seven-fold cover of the sporadic group of Rudvalis. We provide explicit expressions for all the McKay-Thompson series arising.