Ordered Spanning Sets for Vertex Operator Algebras and their Modules

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.