Modular Products and Modules for Finite Groups

TL;DR

This paper introduces a new construction linking different types of moonshine phenomena using singular theta lifts and module theory, providing a foundation for further exploration of penumbral and monstrous moonshine relationships.

Contribution

It develops a general method to connect weight one-half and weight zero moonshine through modules and theta lifts, advancing understanding of moonshine's algebraic structures.

Findings

Establishes a construction translating between moonshine types

Provides a framework for analyzing penumbral and monstrous moonshine

Lays groundwork for a detailed study of moonshine connections

Abstract

Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in weight zero. This construction serves as a foundation for a companion paper in which we explore the connection between penumbral Thompson moonshine and a special case of generalized monstrous moonshine in detail.