Abstract: In the 1970s, as mathematicians worked to classify the finite simple groups, Ogg, McKay and others observed several striking apparent coincidences connecting the then-conjectural Monster group (the largest of the sporadic simple groups) to the theory of modular functions. These ‘coincidences’ became known as “Monstrous Moonshine” and were made into a precise conjecture by Conway and Norton. They conjectured the existence of a naturally occurring graded infinite dimensional Monster module whose graded traces at elements of the monster group give the Fourier coefficients of distinguished modular functions. Borcherds proved the conjecture in 1992, embedding Moonshine in a deeper theory of vertex operator algebras. For this work Borcherds was awarded a Fields Medal. Fifteen years after Borcherds’ proof, Witten conjectured important connections between Monstrous Moonshine and pure quantum gravity in three dimensions. Under Witten’s theory, the irreducible components of the Monster module represent black hole states. Witten asked how these states are distributed. In joint work with Ken Ono and John Duncan, we answer Witten’s question giving exact formulas for these distributions.
Moonshine type-phenomena have been observed for other groups besides the Monster. Notably, the Umbral Moonshine conjectures of Cheng, Duncan, and Harvey connects the automorphism groups of the 24 Niemeier lattices to the Fourier coefficients of certain mock modular forms. Many mathematical physicists anticipate physical interpretations for Umbral Moonshine similar to Witten’s application of Monstrous Moonshine. The first case of Umbral Moonshine, connected to the Leech Lattice, is covered by Monstrous Moonshine, while the second is covered by Gannon’s proof in 2013 of Moonshine for the Mathieu group M24. We verify the remaining 22 cases.