Generalised Umbral Moonshine

Journal Article · Fri Mar 01 23:00:00 EST 2019 · Symmetry, Integrability and Geometry: Methods and Applications

DOI:https://doi.org/10.3842/sigma.2019.014· OSTI ID:1507154

Cheng, Miranda C. N. ^[1]; de Lange, Paul ^[2]; Whalen, Daniel P. Z. ^[3]

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1. Korteweg-de Vries Institute for Mathematics, Amsterdam (The Netherlands); Univ. of Amsterdam, Amsterdam (The Netherlands)
2. Univ. of Kentucky, Lexington, KY (United States)
3. Stanford Univ., Stanford, CA (United States)

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 study we introduce the notion of generalized umbral moonshine, which includes the generalized Mathieu moonshine 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 H³(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 generalized umbral moonshine. We also discuss the possible origin of the generalized umbral moonshine.

• Research Organization: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)
• Sponsoring Organization: USDOE
• Contributing Organization: Korteweg-de Vries Institute for Mathematics, The Netherlands; University of Kentucky, USA; Stanford University, USA
• Grant/Contract Number: AC02-76SF00515
• OSTI ID: 1507154
• Journal Information: Symmetry, Integrability and Geometry: Methods and Applications, Journal Name: Symmetry, Integrability and Geometry: Methods and Applications Journal Issue: 14 Vol. 15; ISSN 1815-0659
• Publisher: Institute of Mathematics, National Academy of Sciences Ukraine
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
finite group representations
group cohomology
mock modular form
moonshine