Mathieu Moonshine in the elliptic genus of K3

TL;DR

This paper provides explicit formulas and extensive evidence supporting the conjecture that the elliptic genus of K3 surfaces encodes dimensions of Mathieu group M24 representations, revealing a deep connection between geometry, modular forms, and group theory.

Contribution

It derives explicit formulas for all twining genera of the K3 elliptic genus and confirms the Mathieu moonshine conjecture through detailed coefficient analysis.

Findings

Explicit formulas for all twining genera.

Verification of non-negative integer multiplicities for the first 500 coefficients.

Strong evidence supporting the Mathieu moonshine conjecture.

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.