Generalised Umbral Moonshine

TL;DR

This paper introduces the concept of generalised umbral moonshine, extending the known relations between finite groups and mock modular forms, and provides supporting data and conjectures about the underlying algebraic structures involved.

Contribution

It defines generalised umbral moonshine, connects it to deformed Drinfel'd doubles of finite groups, and proposes conjectural rules for associating modules to these groups.

Findings

Supporting data for generalised umbral moonshine.

Conjecture on assigning modules to deformed Drinfel'd doubles.

Extension of Mathieu moonshine to a broader framework.

Abstract

Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon which connects finite groups and distinguished modular objects. In this paper we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfel'd (or quantum) double of each umbral finite group $G$, specified by a cohomology class in $H^3(G,U(1))$. We conjecture that in each of the 23 cases there exists a rule to assign an infinite-dimensional module for the deformed Drinfel'd double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine.