A twist in the M24 moonshine story

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.

39 pages, 2 figures; some explanations and clarifications added; accepted for publication by Confluentes Mathematici

References in corpus (4)

• Notes on the K3 Surface and the Mathieu group M_24
• Note on Twisted Elliptic Genus of K3 Surface
• Mathieu Moonshine in the elliptic genus of K3
• Snapshots of Conformal Field Theory

Cited by in corpus (13)

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• Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory
• Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
• Hodge-elliptic genera and how they govern K3 theories
• Snapshots of Conformal Field Theory
• Mathieu Moonshine and Symmetry Surfing
• Not Doomed to Fail
• SU(2) channels the cancellation of K3 BPS states
• Vertex operator superalgebra/sigma model correspondences: The four-torus case
• Tracking the symmetries of ${\mathbb{Z}}_3$-orbifold K3s within the Mathieu groups