Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$
Gerald Höhn, Geoffrey Mason
TL;DR
This work classifies finite groups G that can act as symplectic automorphisms on hyperkähler manifolds deformation equivalent to K3^{[2]}. It reduces the problem to lattice-theoretic constraints via an embedding into the Leech lattice and the Conway group Co_0, yielding two main families: subgroups of M_{23} with at least four orbits on 24 points, and subgroups of Co_0 tied to S-lattices (notably 3^{1+4}.2.2^2 and 3^4.A_6). The authors enumerate 198 admissible Con0_0-conjugacy classes, identify 69 coinvariant lattices L_G, and show how many realize symplectic actions on K3^{[2]}—leading to at least 243 deformation classes of such actions. They further connect these geometric symmetries to Mathieu Moonshine, showing that the equivariant elliptic genus and the second quantized genus for several Moonshine classes match Moonshine predictions, thereby linking hyperkähler geometry, lattice theory, and Moonshine phenomena. Overall, the paper extends Mukai’s K3-surface classification to the K3^{[2]} setting, provides explicit realizations and deformation classifications, and reveals deep ties to moonshine via refined elliptic-genus invariants.
Abstract
We determine the possible finite groups $G$ of symplectic automorphisms of hyperkähler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having at least four orbits in its natural permutation representation on $24$ elements, or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to $\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$ which are maximal with respect to these properties, and (in most cases) we determine all deformation equivalence classes of such group actions. We also compare our results with the predictions of Mathieu Moonshine.