An Overview of Penumbral Moonshine

TL;DR

This paper introduces penumbral moonshine, a new class of relationships between finite groups and modular forms, expanding the framework of moonshine phenomena beyond Mathieu and Thompson moonshine.

Contribution

It defines and explores the concept of penumbral moonshine, generalizing existing moonshine theories to a broader context involving finite groups and vector-valued modular forms.

Findings

Penumbral moonshine links finite groups with vector-valued modular forms.

It generalizes Mathieu and Thompson moonshine phenomena.

The paper explains key features of this new moonshine class.

Abstract

As Mathieu moonshine is a special case of umbral moonshine, Thompson moonshine (in half-integral weight) is a special case of a family of similar relationships between finite groups and vector-valued modular forms of a certain kind. We call this penumbral moonshine. We introduce and explain some features of this phenomenon in this work.