Hyperk\"ahler Isometries Of K3 Surfaces

TL;DR

This paper classifies the possible hyperk"ahler isometry groups of K3 surfaces, identifying 40 groups all realizable in the moduli space and subgroups of the Mathieu group M23, filling a gap in the literature.

Contribution

It provides a complete classification of hyperk"ahler isometry groups of K3 surfaces, expanding understanding of their symmetry groups.

Findings

Explicit list of 40 possible groups

All groups are subgroups of M23

All groups are realized in the moduli space

Abstract

We consider symmetries of K3 manifolds. Holomorphic symplectic automorphisms of K3 surfaces have been classified, and observed to be subgroups of the Mathieu group $M_{23}$. More recently, automorphisms of K3 sigma models commuting with $SU(2)\times SU(2)$ $R$-symmetry have been classified by Gaberdiel, Hohenegger, and Volpato. These groups are all subgroups of the Conway group. We fill in a small gap in the literature and classify the possible hyperk\"ahler isometry groups of K3 manifolds. There is an explicit list of $40$ possible groups, all of which are realized in the moduli space. The groups are all subgroups of $M_{23}$.