Communications in Number Theory and Physics

Volume 4 (2010)

Number 4

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

Pages: 623 – 657



Miranda C.N. Cheng (Department of Mathematics, Harvard University)


A close relationship between$K3$ surfaces and the Mathieu groups has been establishedin the last century. Furthermore, it has been observedrecently that the elliptic genus of $K3$ has a naturalinterpretation in terms of the dimensions ofrepresentations of the largest Mathieu group $M_{24}$. Inthis paper, we first give further evidence for thispossibility by studying the elliptic genus of $K3$ surfacestwisted by some simple symplectic automorphisms. Thesepartition functions with insertions of elements of $M_{24}$(the McKay–Thompson series) give further information aboutthe relevant representation. We then point out that thisnew “moonshine” for the largest Mathieu group is connectedto an earlier observation on a moonshine of $M_{24}$through the $1/4$-BPS spectrum of $K3\timesT^2$-compactified type II string theory. This insight onthe symmetry of the theory sheds new light on thegeneralized Kac–Moody algebra structure appearing in thespectrum, and leads to predictions for new elliptic generaof $K3$, perturbative spectrum of the toroidallycompactified heterotic string, and the index for the$1/4$-BPS dyons in the $d=4$, $\mathcal{N}=4$ stringtheory, twisted by elements of the group of stringy $K3$isometries.

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