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Notes on the K3 Surface and the Mathieu group M_24
Tohru Eguchi, Hirosi Ooguri, Yuji Tachikawa
TL;DR
The paper addresses the unexplained relationship between K3 geometry and M24 by recalling the lattice and coding-theoretic construction of M24 and analyzing K3 symmetry groups. It shows that K3 symmetries embed into restricted M24 subgroups, while suitable M24 subgroups can correspond to K3 symmetries.
Problem
The close connection between M24 and K3 geometry, including the relation between K3 symmetries and M24 subgroups, requires explanation.
Method
The paper constructs M24 from an even self-dual lattice and the extended binary Golay code, then relates K3 symmetry groups to lattice complements and the global Torelli theorem.
Results
K3 symmetry groups are subgroups of M24, cannot be M24 itself, and correspond conversely to M24 subgroups whose 24-point action has at least five orbits.
Takeaways & Limitations
The K3–M24 relationship is constrained by the even self-dual K3 cohomology lattice and by the orbit structure of M24 subgroups.
Abstract
from arXiv · showhide
We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M_24. The reason is yet a mystery.
A Data of M24
M24 is characterized through its 24 conjugacy classes and irreducible representations, whose dimensions include complex-conjugate pairs and an additional real 1035-dimensional representation. Table 1 presents its character table.
- M24 has 26 conjugacy classes and 26 irreducible representations.
- Some irreducible representations occur in complex conjugate pairs.
- M24 also has an extra real irreducible representation of dimension 1035.
- Table 1 gives the character table of M24.
B M24 and the classical geometry of K3
The appendix constructs M24 from the extended binary Golay code and relates it to K3 geometry through the even self-dual cohomology lattice. K3 symmetries form restricted subgroups of M24, and suitable M24 subgroups can conversely arise as K3 symmetries.
- B M24 and the classical geometry of K3: M24 is defined as the coordinate-permutation subgroup preserving the uniquely determined extended binary Golay code.The code arises from a 12-dimensional subspace with all weights divisible by 4 and no weight-4 element.
- B M24 and the classical geometry of K3: The K3 cohomology lattice is an even self-dual 24-dimensional lattice with signature (4, 20), explaining its close connection to M24.
- B M24 and the classical geometry of K3: A symmetry group G of a K3 surface preserving the holomorphic 2-form embeds as a subgroup of M24.This follows by placing the preserved-lattice complement inside the relevant lattice substructure using Nikulin’s result.
- B M24 and the classical geometry of K3: G cannot equal M24 because its action on 24 points must split them into at least five orbits.The bound comes from the preserved cohomology classes and Kähler form, which imply a fixed subspace of at least five dimensions.
- B M24 and the classical geometry of K3: Conversely, a subgroup of M24 acting on 24 points with at least five orbits can yield a K3 surface whose symmetry is G.The construction uses the action on H1,1 together with the global Torelli theorem.
- B M24 and the classical geometry of K3: The Fermat quartic has symmetry (Z4)2 ⋊ S4 with 384 elements and orbit lengths 1, 1, 2, 4, and 16.This symmetry is a subgroup of M24 and its 24-point action decomposes into five orbits.