The overarching finite symmetry group of Kummer surfaces in the Mathieu group M_24

TL;DR

This paper constructs a large overarching symmetry group for Kummer surfaces using the Mathieu group M_24, linking symplectic automorphisms across different K3 surfaces and surpassing known automorphism group sizes.

Contribution

It develops a method to unify symmetries of Kummer surfaces into a single large group within M_24, extending lattice embedding techniques for all such surfaces.

Findings

Constructed a bijection between K3 homology lattice and Niemeier lattice.

Generated an overarching symmetry group of order 40320 for Kummer surfaces.

Contained all symplectic automorphisms leaving the Kaehler class invariant.

Abstract

In view of a potential interpretation of the role of the Mathieu group M_24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different 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 (A_1)^24, which is simultaneously compatible with the finite 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 M_24, generating the 'overarching finite symmetry group' (Z_2)^4:A_7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest finite 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 Kaehler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that finite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M_23, and we extend their techniques of lattice embeddings for all Kummer surfaces with Kaehler class induced from the underlying torus.