The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24

Abstract

In view of a potential interpretation of the role of the Mathieu group M24 in the context of strings compactied on K3 surfaces, we develop techniques to combine groups of symmetries from dierent K3 surfaces to larger `overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type A24 1 , which is simultaneously compatible with the nite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice serves to express all these symplectic automorphisms as elements of the Mathieu group M24, generating the `overarching nite symmetry group' (Z2)4oA7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest nite symplectic automorphism group of a K3 surface by orders of magnitude. For every Kummer surface this group contains the group of symplectic automorphisms leaving the Kahler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that nite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M23, and we extend their techniques of lattice embeddings for all Kummer surfaces with Kahler class induced from the underlying torus.