Rational K3 Homotopy and the Largest Mathieu Group

We interpret the ranks of the rational homotopy groups of a K3 surface as dimensions of representations for the largest sporadic simple Mathieu group. We then construct a vertex algebra equipped with an action by the largest Mathieu group, and use it to associate Jacobi forms to this interpretation, in a compatible way. Our results suggest a topological role for the sporadic simple Mathieu groups in the theory of K3 surfaces.