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A twist in the M24 moonshine story

Taormina, A.; Wendland, K.


K. Wendland


Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z2-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z2-orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3. The 45-dimensional irreducible representation of M24 exhibits a twist, which we prove can be undone in the case of Z2-orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group Z2^4 : A8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.


Taormina, A., & Wendland, K. (2015). A twist in the M24 moonshine story. Confluentes mathematici, 7(1), 83-113.

Journal Article Type Article
Acceptance Date Nov 20, 2014
Online Publication Date Oct 22, 2015
Publication Date Oct 22, 2015
Deposit Date Apr 14, 2013
Journal Confluentes Mathematici
Print ISSN 1793-7442
Electronic ISSN 1793-7434
Publisher Institut Camille Jordan
Peer Reviewed Peer Reviewed
Volume 7
Issue 1
Pages 83-113
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