A Nonmeromorphic Extension of the Moonshine Module Vertex Operator Algebra

We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator superalgebra and a twisted module for this vertex operator superalgebra with respect to the involution which is the identity on the even subspace and is −1 on the odd subspace. It also gives the superconformal structures observed by Dixon, Ginsparg and Harvey. The relation between the modular function J(q) = j(q)−744 and dimensions of certain representations of the Monster was first noticed by McKay and Thompson (see [Th]). Based on these observations, McKay and Thompson conjectured the existence of a natural (Z-graded) infinitedimensional representation of the Monster group such that its graded dimension as a Z-graded vector space is equal to J(q). In the famous paper [CN] by Conway and Norton, remarkable numerology between McKay-Thompson series (graded traces of elements of the Monster acting on the conjectured infinite-dimensional representation of the Monster) and modular functions was collected, and surprising conjectures about those modular functions occured were presented. The Monster was constructed by Griess [G] later but the mysterious connection between the Monster and modular functions was still not expained. In [FLM1], Frenkel, Lepowsky and Meurman constructed a natural 1991 Mathematics Subject Classification. Primary 17B69; Secondary 17B68, 81T40. This research is supported in part by NSF grant DMS-9301020 and by DIMACS, an NSF Science and Technology Center funded under contract STC-88-09648.