Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-14T06:01:35.408Z Has data issue: false hasContentIssue false

Rational Vertex Operator Algebras and their Orbifolds

Published online by Cambridge University Press:  06 July 2010

Geoffrey Mason
Affiliation:
Department of Mathematics, UC Santa Cruz
James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
Get access

Summary

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.

Type
Chapter
Information
Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 270 - 281
Publisher: Cambridge University Press
Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×