A twist in the M 24 moonshine story
Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 83-113.

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 2 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a 2 -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 M 24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.

The 45-dimensional irreducible representation of M 24 exhibits a twist, which we prove can be undone in the case of 2 -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group ( 2 ) 4 A 8 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.

DOI: 10.5802/cml.19
Classification: 81T40, 81T60, 14J28

Anne Taormina 1; Katrin Wendland 2

1 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, Durham DH1 3LE, U.K.
2 Mathematics Institute, University of Freiburg, D-79104 Freiburg, Germany
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Anne Taormina; Katrin Wendland. A twist in the $M_{24}$ moonshine story. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 83-113. doi : 10.5802/cml.19. https://cml.centre-mersenne.org/articles/10.5802/cml.19/

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