Monster and Categorification

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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.

This entire process is akin to the “categorification” of Hauptmoduls of the genus 0 surface. To the higher representation theory community, categorification has always been a big task to carry out. Informally, categorification can be understood as associating a natural number to a vector space/module by taking its dimension. It is the positivity of the coefficients of a certain functions or the entries of matrices, alluring us to ponder that there might exist higher structures (vector spaces, modules, categories) governing or functioning in the background, waiting to be discovered (by you).