Mathieu moonshine and \mathcal{N}=2 string compactifications

There is a 'Mathieu moonshine' relating the elliptic genus of K3 to the spo- radic group M24. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3×T 2 to type IIA strings compact- ified on Calabi-Yau threefolds. We demonstrate that dimensions of M24 representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov-Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces Fn.