A twist in the M24 moonshine story

TL;DR

This paper explores the Mathieu Moonshine phenomenon in K3 surface conformal field theories, identifying special 45-dimensional state spaces linked to the Mathieu group M24 and analyzing their symmetry representations.

Contribution

It explicitly constructs 45-dimensional state spaces in K3 CFTs related to Mathieu Moonshine and investigates the twisting of M24 representations under various symmetry groups.

Findings

Identified 45-dimensional state spaces in Z2-orbifold CFTs on K3.

Proved the twist in M24 representations can be undone for geometric symmetry groups.

Conjectured that untwisted representations correspond to geometric symmetry groups.

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.