Mathieu twining characters for K3

TL;DR

The paper tests the proposal that the K3 elliptic genus carries an action of the Mathieu group $\mathbb{M}_{24}$ by constructing twining genera $\phi_g(\tau,z)$ and analyzing their NS-sector characters $\chi_g(\tau)$. It demonstrates that these NS-twining characters have strong modular constraints (invariance under $\Gamma^g$) and, for several small-order conjugacy classes, can be identified with shifted McKay–Thompson series, supporting a Mathieu Moonshine-like structure. Moreover, replication identities linking twining characters across symmetric powers are established, yielding nontrivial consistency checks and refining coefficient identifications relative to earlier work. While not all twining characters have closed forms, the results significantly bolster the view that the K3 elliptic genus encodes a genuine $\mathbb{M}_{24}$ symmetry, with connections to BPS algebras and potential broader Moonshine-like phenomena.

Abstract

The analogue of the McKay-Thompson series for the proposed Mathieu group action on the elliptic genus of K3 is analysed. The corresponding NS-sector twining characters have good modular properties and satisfy remarkable replication identities. These observations provide strong support for the conjecture that the elliptic genus of K3 carries indeed an action of the Mathieu group M24.