# Monster and Categorification

This is a survey/sharing seminar. The Monster group $\mathbb{M}$, being the largest simple (sporadic) finite groups, has intriguing connections to Algebra(Representation Theory, VOAs, etc.), Number Theory (Modular Forms, Hauptmoduls, etc.), and Mathematical Physics (String Theory, CFT, etc.). The history of finding this gigantic group itself is interesting, but the fact that it serves as a bridge between the three aforementioned subjects is even more astounding. At first, there was just a moonshine correlation between a $j$-function and $\mathbb{M}$ coming from McKay’s observation that its first coefficient 196884 $\approx$ 196883. After the Monstrous Moonshine conjecture had been formulated by Conway and Norton, Frenkel-Lepowsky-Meurman created the infinite-dimensional “natural” representation, called the Moonshine module for its automorphism group being the Monster $\mathbb{M}$. This, in turn, provided a playing ground for Borcherds to prove that the graded character of the Moonshine modules are indeed the Hauptmoduls of a genus 0 surface $\mathbb{H} / SL_2(\mathbb{Z})$ by interpreting the Moonshine module as a vertex operator algebra which earned him a Field medal later.