Mathieu Moonshine in the elliptic genus of K3

TL;DR

The paper tackles the problem of whether the K3 elliptic genus exhibits Mathieu Moonshine by constructing the full set of twining genera $\phi_g(\tau,z)$ for all conjugacy classes of $\mathbb{M}_{24}$ and proving their modular properties as Jacobi forms of index $1$ and weight $0$ on appropriate subgroups $\Gamma_0(N)$ with a multiplier system. It develops two complementary methods to determine these twining genera and uses them to obtain explicit closed forms, enabling the decomposition of the elliptic genus coefficients into traces $\mathrm{Tr}_{H_n}(g)$ and hence the $\mathbb{M}_{24}$-decomposition of the $H_n$. The authors verify, up to $n\le 500$ and for $n\le 30$ coefficients, that all multiplicities are non-negative integers, providing strong evidence for the $\mathbb{M}_{24}$-action on BPS states contributing to the K3 elliptic genus. They also discuss implications for deeper symmetry structures beyond $\mathbb{M}_{24}$ and note that not all twining genera are Hauptmoduls, highlighting interesting connections between moonshine, modular forms, and string-theoretic symmetries.

Abstract

It has recently been conjectured that the elliptic genus of K3 can be written in terms of dimensions of Mathieu group M24 representations. Some further evidence for this idea was subsequently found by studying the twining genera that are obtained from the elliptic genus upon replacing dimensions of Mathieu group representations by their characters. In this paper we find explicit formulae for all (remaining) twining genera by making an educated guess for their general modular properties. This allows us to identify the decomposition of all expansion coefficients in terms of dimensions of M24-representations. For the first 500 coefficients we verify that the multiplicities with which these representations appear are indeed all non-negative integers. This represents very compelling evidence in favour of the conjecture.

Theorems & Definitions (1)

• Conjecture