Special Math/Physics Seminar

K3 surfaces, moonshine, and string theory

Sarah Harrison

Harvard University

Moonshine is a mysterious relationship in mathematics between finite groups and modular forms, which appears to have deep connections to physics and string theory. The most famous example is monstrous moonshine, which relates the coefficients of the modular J-function to dimensions of representations of the largest sporadic simple group, the monster group.

I will discuss “umbral moonshine,” a new moonshine phenomenon which relates mock modular forms to automorphism groups of Niemeier lattices, the first case of which was first discovered as a connection between the elliptic genus of a K3 surface and the Mathieu group M24. I will explain how moonshine has connections to symmetries of sigma models arising from string compactification on K3 surfaces, and thus many interesting applications in mathematics and physics, including number theory, enumerative geometry, supersymmetric black holes, string dualities, and AdS/CFT, among other things.