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Some simple extensions of Mathieu Moonshine

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Mathieu Moonshine is a striking and unexpected relationship between the
sporadic simple finite group M24 and a special Jacobi form, the elliptic
genus, which arises naturally in studies of nonlinear sigma models with
K3 target.  In this talk, we first discuss its predecessor (Monstrous
Moonshine), then
discuss the current evidence in favor of Mathieu Moonshine.  We also
discuss extensions of this story involving `second quantized mirror
symmetry,' relating heterotic strings on K3 to type II strings on
Calabi-Yau threefolds.