Much ado about Mathieu

Understand

Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II string theory on K3 surfaces appears to possess a Moonshine for the largest Mathieu group.

• Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24.
• In this paper we prove that the resulting sequence of class functions are true characters of M24, proving the Eguchi-Ooguri-Tachikawa conjecture.
• We prove the evenness property of the multiplicities, as conjectured by several authors.