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

TL;DR

This paper explores the deep connection between K3 surfaces, the Mathieu group M24, and string theory, providing new evidence for Mathieu moonshine through twisted elliptic genera and linking it to BPS spectra and algebraic structures.

Contribution

It offers new evidence for Mathieu moonshine by studying twisted elliptic genera and connects this to string theory spectra and algebraic structures.

Findings

Twisted elliptic genera reveal new representation data of M24.

Connections established between moonshine phenomena and string theory BPS spectra.

Predictions made for elliptic genera and dyon indices in string compactifications.

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.