K3 Surfaces, N=4 Dyons, and the Mathieu Group M24

TL;DR

This work argues for a deep, two-branch moonshine linking K3 geometry and the Mathieu group $M_{24}$ to string theory spectra. By analyzing twisted K3 elliptic genera, it connects McKay–Thompson series $T_g( au)$ to the $M_{24}$-module $K^{ atural}$ and demonstrates how these structures extend to heterotic perturbative spectra via $ ext{eta}_g( au)$ and to $1/4$-BPS dyons through a generalised Kac–Moody algebra. The two moonshine pictures—elliptic-genus based and eta-product based—are shown to cohere in the twisted denominators $rac{1}{oldsymbol{ abla}_g}$ of the dyon algebra, with wall-crossing encoded by automorphic products $oldsymbol{ abla}_g$. The results yield concrete predictions for twisted K3 elliptic genera, twisted heterotic spectra, and twisted dyon indices, offering new evidence for a pervasive $M_{24}$ symmetry in $K3$-compactified string theories and pointing to non-geometric symmetry realizations.

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 M24. 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 M24 (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 M24 through the 1/4-BPS spectrum of K3xT^2-compactified type II string theory. This insight on the symmetry of the theory sheds new light on the generalised 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.