Skip to main content Accessibility help
×
Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-23T03:22:56.176Z Has data issue: false hasContentIssue false

On Certain Automorphic Forms Associated to Rational Vertex Operator Algebras

Published online by Cambridge University Press:  06 July 2010

Antun Milas
Affiliation:
Department of Mathematics, University at Albany, SUNY Albany, NY 12222 E-mail address: amilas@math.albany.edu
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

Abstract

To every rational vertex operator algebra V we associate an automorphic form on Г′ (1) that we call the Wronskian of V. We have previously shown [M2], [M3] that in the case of Virasoro minimal models it is possible to give qualitative arguments about the Wronskian by using the representation theoretic methods. Here we apply the theory of automorphic forms and extend our previous work to a larger class of vertex operator algebras. We also give a detailed analysis of two-dimensional modular invariant spaces that arise from affine Kac-Moody Lie algebras.

As a main byproduct of our analysis we provide new proofs of certain Dyson-Macdonald's identities for powers of the Dedekind η–function for Cl, BCl and Dl series, and related identities (e.g., Jacobi's Four Square Theorem).

Introduction and notation

The existence of a fusion ring and modular invariance of graded dimensions, or characters, are the most interesting features of every rational conformal field theory [MS]. When it comes to vertex operator algebra theory, proving modular invariance [Zh] (cf. [DLM1]) and ultimately the Verlinde formula [Hu1] is a formidable task. A key ingredient in proving modular invariance is played by the so-called C2–cofiniteness [Zh] which, in particular, guarantees the convergence of all one-point functions on the torus. The C2–cofiniteness plays an important role in the proof of the Verlinde conjecture as well [Hu1]. Even though the vector space spanned by irreducible characters is a PSL(2,ℤ)–module (i.e., a modular invariant space), one cannot state the Verlinde formula without having an action of SL(2,ℤ) (the operator S2 does not act as the identity in general–charge conjugation).

Type
Chapter
Information
Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 330 - 357
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
×