K3 surfaces, $\mathcal{N}=4$ dyons and the Mathieu group $M_{24}$

Abstract

A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural interpretation in terms of the dimensions of representations of the largest Mathieu group M 24 . In this paper, we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M 24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new "moonshine" for the largest Mathieu group is connected to an earlier observation on a moonshine of M 24 through the 1/4-BPS spectrum of K3 T 2compactified type II string theory. This insight on the symmetry of the theory sheds new light on the generalized Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spectrum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d = 4, N = 4 string theory, twisted by elements of the group of stringy K3 isometries.