Product Expansions of Conformal Characters

TL;DR

This paper explores infinite series of rational conformal field theories with characters that are modular units, providing a new perspective on their product expansions and proposing an algorithm to verify this property.

Contribution

It introduces new classes of rational conformal field theories with modular unit characters and presents an algorithm to determine whether a modular function is a modular unit.

Findings

Identified infinite series with modular unit characters

Proposed a conjecture on Casimir W-algebra models

Developed an algorithm for testing modular units

Abstract

We describe several infinite series of rational conformal field theories whose conformal characters are modular units, i.e. which are modular functions having no zeros or poles in the upper complex half plane, and which thus possess simple product expansions. We conjecture that certain infinite series of rational models of Casimir W-algebras always have this property. Furthermore, we describe an algorithm which can be used to prove whether a modular function is a modular unit or not.