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Mock modular Mathieu moonshine modules
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  • Published: 28 July 2015

Mock modular Mathieu moonshine modules

  • Miranda C N Cheng1,
  • Xi Dong2,
  • John F R Duncan3,
  • Sarah Harrison2,
  • Shamit Kachru2 &
  • …
  • Timm Wrase2 

Research in the Mathematical Sciences volume 2, Article number: 13 (2015) Cite this article

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Abstract

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of \(\textit{Co}_0\) that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain \({\mathcal {N}}=4\) superconformal algebra. Similarly, any subgroup of \(\textit{Co}_0\) that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain \({\mathcal {N}}=2\) superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the \({\mathcal {N}}=4\) (resp. \({\mathcal {N}}=2\)) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional \({\mathbb {Z}}\)-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.

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Authors and Affiliations

  1. Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    Miranda C N Cheng

  2. Department of Physics and SLAC, Stanford University, Stanford, CA, 94305, USA

    Xi Dong, Sarah Harrison, Shamit Kachru & Timm Wrase

  3. Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH, 44106, USA

    John F R Duncan

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  1. Miranda C N Cheng
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Correspondence to Miranda C N Cheng.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Cheng, M.C.N., Dong, X., Duncan, J.F.R. et al. Mock modular Mathieu moonshine modules. Mathematical Sciences 2, 13 (2015). https://doi.org/10.1186/s40687-015-0034-9

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  • Received: 01 July 2014

  • Accepted: 15 June 2015

  • Published: 28 July 2015

  • DOI: https://doi.org/10.1186/s40687-015-0034-9

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Keywords

  • Elliptic Genus
  • Jacobi Form
  • Superconformal Algebra
  • Sporadic Group
  • Global Symmetry Group
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