Symmetries of K3 sigma models
Matthias R. Gaberdiel, Stefan Hohenegger, Roberto Volpato
TL;DR
This work classifies the supersymmetry-preserving automorphisms of K3 ${\rm N}=(4,4)$ sigma-models and proves they sit inside the Conway group ${\rm Co}_1$, extending Mukai–Kondo from geometry to string theory. It develops a lattice-based argument via embeddings into the Leech lattice and analyzes three concrete CFT realizations—the ${\mathbb T}^4/\mathbb{Z}_2$ orbifold and the Gepner points $(2)^4$ and $(1)^6$—to show the actual symmetry groups can lie outside ${\rm M}_{24}$ while remaining within ${\rm Co}_1$. The paper also details the D-brane charge lattices in these models and computes twining genera, revealing both ${\rm M}_{24}$-consistent and non-Mathieu genera, thereby refining how moonshine phenomena arise in K3 compactifications. Overall, the results provide a concrete stringy counterpart to Mukai’s theorem and illuminate how moduli-space symmetries manifest in physical theories of K3. The findings have implications for understanding symmetry structures in string compactifications and their connections to sporadic groups and moonshine-like phenomena.
Abstract
It is shown that the supersymmetry-preserving automorphisms of any non-linear sigma-model on K3 generate a subgroup of the Conway group Co_1. This is the stringy generalisation of the classical theorem, due to Mukai and Kondo, showing that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M_{23}. The Conway group Co_1 contains the Mathieu group M_{24} (and therefore in particular M_{23}) as a subgroup. We confirm the predictions of the Theorem with three explicit CFT realisations of K3: the T^4/Z_2 orbifold at the self-dual point, and the two Gepner models (2)^4 and (1)^6. In each case we demonstrate that their symmetries do not form a subgroup of M_{24}, but lie inside Co_1 as predicted by our Theorem.