Symmetry-surfing the moduli space of Kummer K3s

Abstract

A maximal subgroup of the Mathieu group M24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kaehler class is induced from the underlying complex torus. As a subgroup of M24, this group is the stabilizer group of an octad in the Golay code. To meaningfully combine the symmetry groups of distinct Kummer surfaces, we introduce the concepts of Niemeier markings and overarching maps between pairs of Kummer surfaces. The latter induce a prescription for symmetry-surfing the moduli space, while the former can be seen as a first step towards constructing a vertex algebra that governs the elliptic genus of K3 in an M24-compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit.

20 pages; minor changes; accepted for publication in the Proceedings Volume of String-Math 2012

References in corpus (4)

• Note on Twisted Elliptic Genus of K3 Surface
• Chiral de Rham complex and the half-twisted sigma-model
• K3 Surfaces, N=4 Dyons, and the Mathieu Group M24
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• An Uplifting Discussion of T-Duality
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• Eta Products, BPS States and K3 Surfaces
• Comments on M${}_{24}$ representations and $C{Y}_3$ geometries
• Hodge-elliptic genera and how they govern K3 theories
• Snapshots of Conformal Field Theory
• Second Quantized Mathieu Moonshine
• On Mathieu moonshine and Gromov-Witten invariants
• SU(2) channels the cancellation of K3 BPS states
• Not Doomed to Fail
• Calabi-Yau manifolds and sporadic groups
• Tracking the symmetries of ${\mathbb{Z}}_3$-orbifold K3s within the Mathieu groups