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Ordered Spanning Sets for Vertex Operator Algebras and their Modules

Published online by Cambridge University Press:  06 July 2010

Geoffrey Buhl
Affiliation:
Mathematics Department Rutgers University Piscataway, NJ 08854-8019 gbuhl@math.rutgers.edu
James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
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Summary

Abstract

Moonshine relates three fundamental mathematical objects: the Monster sporadic simple group, the modular function j(τ), and the moonshine module vertex operator algebra V♭. Examining the relationship between modular functions and the representation theory of vertex operator algebras reveals rich structure. In particular, C2-cofiniteness (also called Zhu's finiteness condition) implies the existence of finite generating sets and Poincaré-Birkhoff-Witt-like spanning sets for vertex operator algebras and their modules. These spanning sets feature desirable ordering restrictions, e.g., a difference-one condition.

Introduction

The theory of vertex operator algebra blossomed from two major accomplishments: the proof of the McKay-Thompson conjecture by Frenkel, Lepowsky, and Meurman [FLM88] who constructed the Moonshine module V and the proof of the Conway-Norton conjecture by Borcherds [Bor92] using the Moonshine module. These two conjectures make up what is commonly referred to as Monstrous Moonshine, relating the modular function j(τ) and the Monster group by way of a third fundamental mathematical object, the Moonshine module vertex operator algebra V. The study of vertex operator algebras continues to reveal relations within mathematics and with physics.

Representation theory is a particularly rich aspect of the theory of vertex operator algebras with fundamental connections to number theory, the theory of simple groups, and string and conformal field theories in physics. A core idea in the representation theory of vertex operator algebras and conformal field theory is “rationality”, a term used in a variety of ways to describe certain desirable properties of a vertex operator algebra and its modules.

Type
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Information
Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 40 - 54
Publisher: Cambridge University Press
Print publication year: 2010

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