Introduction

This paper is based on a talk I gave at the Spitalfields Day, held under the auspices of the London Mathematical Society during the Edinburgh Conference on Moonshine. As such, it is intended for nonspecialists.

Consider the following confluence of ideas:

• As closed strings move in space-time they sweep out a worldsheet which carries the structure of a Riemann surface.

• Riemann surfaces occur as quotients Г \ H of the complex upper half-plane H by arithmetic groups Г ⊆ SL(2,R).

• The modular forms associated to Г \ H carry arithmetic information manifested in the coefficients of the corresponding Fourier series.

• Vertex operator algebra theory may be construed as an algebraicization of aspects of the theory of elementary particles (bosonic strings) and their interactions.

• Monstrous Moonshine relates certain distinguished elliptic modular functions (so-called hauptmoduln) to the Monster simple group M.

• M is the automorphism group of a particular vertex operator algebra, the Moonshine Module V♭, which models bosons in the critical dimension c = 24 as a Z2-orbifold.

The reader who is not conversant with all of these ideas should have no fear. The statements above are intended only as background, and to suggest several points. The first is the inevitability of connections between elliptic modular forms, vertex operator algebras and finite groups; the second is the intimate connection with ideas from theoretical physics (string theory and conformal field theory).

Introduction

The Conway-Norton conjectures about the Monster-modular connection, later established by Borcherds [B] following the work of Frenkel-Lepowsky-Meurman [FLM], set the stage for an intensive study of the origins of the relations between finite groups and modular functions. Norton also introduced generalized moonshine which associates q-expansions to pairs of commuting elements in M. His conjecture that the nonconstant functions that one obtains in this way are also hauptmoduln remains open.

By now it is clear that the principal mathematical idea underlying the general study of such phenomena is that of a vertex operator algebra. For an extensive class of vertex operator algebras, so-called rational orbifold models, one expects a theory completely parallel to Monstrous Moonshine whereby one associates modular functions to automorphisms of finite order. Furthermore, this theory naturally accommodates generalized moonshine. From this perspective, the Conway-Norton phenomena would be a particularly interesting spradic example of a general theory, just as the Monster itself is a particularly interesting sporadic example of a finite simple group.

In the spirit of the Edinburgh Conference, this paper is mainly devoted to a review of some of the main results currently available concerning the structure of rational vertex operator algebras and their orbifolds. We sometimes use the Frenkel-Lepowsky-Meurman Moonshine Module V♭ [FLM] to illustrate the ideas. We also suggest open problems - many of them well-known - whose solution would contribute to a more complete theory.

Abstract. We discuss the foundations of vertex operator algebras and their representations, concentrating on rational vertex operator algebras. We use the Witt-Grothendieck group of conformal field theories as a vehicle to describe results, in particular the connections with modular forms.

INTRODUCTION

Vertex operator algebras suggest themselves as objects that could play a rôle in a geometric description of elliptic cohomology and related topics. As linear spaces with lots of symmetry they can participate in K-theoretic type constructions, and many (but not all) vertex operator algebras are endowed with a natural modular form as part of their structure. The incorporation of vertex operator algebras into topology is well under way (e.g. [Bo], [MS], [MSV], [T]), but it seems true to say that the underlying algebraic theory is not well understood by many potential users of the subject. What follows is an attempt to convey some of the basic ideas about vertex operator algebras, in particular the rapidly advancing theory of rational vertex operator algebras. These are the algebras most naturally associated to modular forms, and in the guise of RCFT (rational conformal field theory) they constitute an active area of research in physics ([FMS]) as well as mathematics. For more information and background the reader may refer to the following: [DM4], [FLM], [G], [K1], [KR], [QFS]. Additional references will be mentioned below.

In my talk at the Newton Institute I emphasized questions about group actions (orbifold theory) and in particular how one can get information about finite group cohomology (e.g., for the Monster group) by looking at maps of equivariant Witt-Grothendieck groups of vertex operator algebras into group cohomology.

Introduction. The celebrated paper of Conway-Norton on connections between the Monster and certain modular functions opened up dramatic possibilities for the study of hitherto unrelated fields, but for a long time the foundations of this new subject remained obscure. Work of Borcherds and Frenkel-Lepowsky-Meurman showed that the origins of “moonshine” involved the theory of infinite-dimensional Lie algebras and that indeed the monster-modular connection could be understood in the context of two-dimensional conformal field theory. In an appendix to, Norton extended the Conway-Norton conjectures in a quite remarkable way. He expected that his new strengthened conjectures, as compelling as they were, were peculiar to the Monstrous situation and might help lead to an explanation of the genus zero problem. But these hopes were quickly dashed, in the sense that in I showed that, roughly speaking, one can associate modular forms to the elements of any finite group in such a way that the axioms which Norton introduced were satisfied. Shortly thereafter I learned that the ideas of Norton were just those which axiomatize certain algebraic aspects of “conformal field theory on an orbifold.”

Thus indeed Norton's ideas are susceptible to application to any finite group and it is my feeling that this approach will provide a suitable foundation to the subject of moonshine. In this note I want to discuss the notion of what I call an “elliptic system,” which is just a mild extension of Norton's ideas in, and show how it can be thought of as a simultaneous generalization of a modular form and of a “moonshine module.”