A twist in the M24 moonshine story

Taormina, A.; Wendland, K.

Abstract

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.

Citation

Taormina, A., & Wendland, K. (2015). A twist in the M24 moonshine story. Confluentes mathematici, 7(1), 83-113. https://doi.org/10.5802/cml.19

• 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
• DOI: https://doi.org/10.5802/cml.19
• Public URL: https://durham-repository.worktribe.com/output/1456309
• Related Public URLs: http://arxiv.org/abs/1303.3221