Vertex Operators and Modular Forms

TL;DR

This paper explores the connection between vertex operator algebras and modular forms, illustrating how algebraic structures influence arithmetic invariants and vice versa, with examples.

Contribution

It explains the relationship between VOAs and modular forms, providing insights and examples of how algebraic and arithmetic properties interact.

Findings

VOAs relate to elliptic functions and modular forms

Algebraic structures impose restrictions on invariants

Examples illustrate the interplay between algebra and arithmetic

Abstract

The leitmotif of these Notes is the idea of a vertex operator algebra (VOA) and the relationship between VOAs and elliptic functions and modular forms. This is to some extent analogous to the relationship between a finite group and its irreducible characters; the algebraic structure determines a set of numerical invariants, and arithmetic properties of the invariants provides feedback in the form of restrictions on the algebraic structure. One of the main points of these Notes is to explain how this works, and to give some reasonably interesting examples.