Character Vectors of Strongly Regular Vertex Operator Algebras

TL;DR

This paper explores the relationship between vertex operator algebras and number theory, focusing on modular forms, Zhu's theorem, and classification results, including new examples and potential character vectors for VOAs.

Contribution

It provides a new perspective on Zhu's theorem, axiomatizes properties of modular forms relevant to VOAs, and presents new potential character vectors in higher ranks.

Findings

Summarizes known classification results in rank two.

Introduces new examples of potential character vectors.

Highlights open questions about the existence of certain VOAs.

Abstract

We summarize interactions between vertex operator algebras and number theory through the lens of Zhu theory. The paper begins by recalling basic facts on vertex operator algebras (VOAs) and modular forms, and then explains Zhu's theorem on characters of VOAs in a slightly new form. We then axiomatize the desirable properties of modular forms that have played a role in Zhu's theorem and related classification results of VOAs. After this we summarize known classification results in rank two, emphasizing the geometric theory of vector-valued modular forms as a means for simplifying the discussion. We conclude by summarizing some known examples, and by providing some new examples, in higher ranks. In particular, the paper contains a number of potential character vectors that could plausibly correspond to a VOA, but such that the existence of a corresponding hypothetical VOA is presently unknown.