Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 014, 27 pages      arXiv:1608.07835      https://doi.org/10.3842/SIGMA.2019.014
Contribution to the Special Issue on Moonshine and String Theory

Generalised Umbral Moonshine

Miranda C.N. Cheng ab, Paul de Lange c and Daniel P.Z. Whalen d
a) Korteweg-de Vries Institute for Mathematics, Amsterdam, The Netherlands
b) Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands
c) Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
d) Stanford Institute for Theoretical Physics, Department of Physics and Theory Group, SLAC, Stanford University, Stanford, CA 94305, USA

Received October 08, 2018, in final form January 30, 2019; Published online March 02, 2019

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.

Key words: moonshine; mock modular form; finite group representations; group cohomology.

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