The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24

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 24 1 , 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 Kähler 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 Kähler class induced from the underlying torus.

Calabi-Yau manifolds and sporadic groups

“…These questions are not easily answered but it has been shown that twined elliptic genera for K3 manifolds give all the McKay-Thompson series of Mathieu moonshine plus several extra twined elliptic genera that one would not have expected based on Mathieu moonshine alone [18]. At the same time the idea of symmetry surfing has been pursued, in which one tries to find an explicit M 24 symmetry by combining the symmetry groups at different points in K3 moduli space [13][14][15][16]. In this paper we followed a different path and studied the elliptic genus for other Calabi-Yau manifolds with an eye for potential connections to sporadic groups.…”

A K3 sigma model with $ \mathbb{Z}_2^8 $ : $ {{\mathbb{M}}_{20 }} $ symmetry

“…As it turns out the relevant K3 sigma model can be described as the usual Z 2 -orbifold of a torus theory at the special D 4 -point, such that the bosonic theory before orbifolding has an so(8) 1 current symmetry, both for left-and right-movers. From this geometric viewpoint, the model is a nonlinear sigma model on the so-called tetrahedral Kummer K3 studied in detail in [21].…”

“…12, all of whose boundary conditions are coupled. In appendix B, we state the correspondence in detail in terms of the four leftand four right-moving Dirac fermions 21) which satisfy the OPEs…”

“…The symmetry group of that Kummer surface was studied in detail in [21]; in particular, the group of holomorphic symplectic automorphisms is the group T 192 ∼ = (Z 2 ) 4 : A 4 of order 192. The subgroup of type A 4 of T 192 is induced by those symmetries of the underlying torus that have a fixed point.…”

“…Since the elliptic genus is constant along each connected component of the moduli space of N = (4, 4) superconformal field theories at central charge (c, c) = (6, 6), one may then hope that the symmetries at different points in moduli space may be put together. This led two of us to suggest that an 'overarching symmetry group' based on the classical geometric symmetries of K3 nonlinear sigma models could be defined in this manner [21]; indeed already under restriction to symmetries of Kummer K3 s one obtains the group Z 4 2 : A 8 , which is a maximal subgroup of M 24 not contained in M 23 [22,23].…”