Symmetry-Surfing the Moduli Space of Kummer K3s

A maximal subgroup of the Mathieu group M-24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kahler class is induced from the underlying complex torus. As a subgroup of M-24, 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 M-24-compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit.