Generalised Umbral Moonshine

Cheng, Miranda C. N.; de Lange, Paul; Whalen, Daniel P. Z.

doi:10.3842/sigma.2019.014

Title: Generalised Umbral Moonshine

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 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.

Authors:

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

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

Publication Date:: 2019-03-02

Research Org.:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)

Sponsoring Org.:: USDOE

Contributing Org.:: Korteweg-de Vries Institute for Mathematics, The Netherlands; University of Kentucky, USA; Stanford University, USA

OSTI Identifier:: 1507154

Grant/Contract Number:: AC02-76SF00515

Resource Type:: Accepted Manuscript

Journal Name:: Symmetry, Integrability and Geometry: Methods and Applications

Additional Journal Information:: Journal Volume: 15; Journal Issue: 14; Journal ID: ISSN 1815-0659

Publisher:: Institute of Mathematics, National Academy of Sciences Ukraine

Country of Publication:: United States

Language:: English

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

Citation Formats


                    Cheng, Miranda C. N., de Lange, Paul, and Whalen, Daniel P. Z.. Generalised Umbral Moonshine.  United States: N. p., 2019. 
Web.  doi:10.3842/sigma.2019.014.

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                    Cheng, Miranda C. N., de Lange, Paul, & Whalen, Daniel P. Z.. Generalised Umbral Moonshine.  United States.  https://doi.org/10.3842/sigma.2019.014

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                    Cheng, Miranda C. N., de Lange, Paul, and Whalen, Daniel P. Z.. Sat .  
"Generalised Umbral Moonshine".  United States.  https://doi.org/10.3842/sigma.2019.014.  https://www.osti.gov/servlets/purl/1507154.

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@article{osti_1507154,

  title        = {Generalised Umbral Moonshine},

  author       = {Cheng, Miranda C. N. and de Lange, Paul and Whalen, Daniel P. Z.},

  abstractNote = {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.},

  doi          = {10.3842/sigma.2019.014},

  journal      = {Symmetry, Integrability and Geometry: Methods and Applications},

  number       = 14,

  volume       = 15,

  place        = {United States},

  year         = {2019},

  month        = {3}

}

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Works referenced in this record:

Orbifolds, Hopf algebras, and the Moonshine
journal, July 1991

Bantay, P.
Letters in Mathematical Physics, Vol. 22, Issue 3
DOI: 10.1007/bf00403544

Beauty and the beast: Superconformal symmetry in a monster module
journal, June 1988

Dixon, L.; Ginsparg, P.; Harvey, J.
Communications in Mathematical Physics, Vol. 119, Issue 2
DOI: 10.1007/bf01217740

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Title: Generalised Umbral Moonshine

Abstract

Citation Formats

Orbifolds, Hopf algebras, and the Moonshine journal, July 1991

Beauty and the beast: Superconformal symmetry in a monster module journal, June 1988

Orbifolds, Hopf algebras, and the Moonshine
journal, July 1991

Beauty and the beast: Superconformal symmetry in a monster module
journal, June 1988