Monstrous and Generalized Moonshine and Permutation Orbifolds

Summary

Abstract

We consider the application of permutation orbifold constructions towards a new possible understanding of the genus zero property in Monstrous and Generalized Moonshine. We describe a theory of twisted Hecke operators in this setting and conjecture on the form of Generalized Moonshine replication formulas.

Introduction

The Conway and Norton Monstrous Moonshine Conjectures [CN], the construction of the Moonshine Module [FLM] as an orbifold vertex operator algebra and the completion of the proof of Monstrous Moonshine by Borcherds [Bo2] provided much of the motivation for the development of Vertex Operator Algebras (VOAs) e.g. [Bo1], [FLM], [Ka], [MN]. Another highlight of VOA theory is Zhu's study of the modular properties of the partition function (and n-point functions) for generic classes of VOAs [Z]. Zhu's ideas were generalized to include orbifold VOAs [DLM] whose relevance to Monstrous Moonshine is emphasized in refs. [T1], [T2]. Norton's Generalized Moonshine Conjectures [N2] concerning centralizers of the Monster group has yet to be generally proven using either Borcherds' approach or orbifold partition function methods although some progress has recently been made in refs. [H] and [T3], [IT1], [IT2] respectively.

In this note, we sketch a possible new approach to these areas based on permutation orbifold VOA constructions [DMVV], [BDM]. In particular, we introduce a theory of twisted Hecke operators generalizing classical Hecke operators in number theory e.g. [Se]. We then discuss permutation orbifold constructions where the classical Hecke operators naturally appear and finite group and permutation orbifold constructions where the twisted Hecke operators appear.