Mathieu moonshine and the geometry of k3 surfaces

Abstract

We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for the Mathieu group M24 and also for a certain vertex operator superalgebra VG where G is the holonomy group.