Andrew O’Desky with John F. R. Duncan
Dynkin Diagrams and the Groups of Umbral Moonshine
Elliptic curves are number theoretic objects with many important applications in mathematics, cryptography, string theory and beyond. The supersingular elliptic curves in prime characteristic constitute a special family, and in 1974 Andrew Ogg showed that there are exactly 15 primes p such that every supersingular elliptic curve in characteristic p can be described using integers; namely, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71. It was soon thereafter that a fantastic coincidence was noticed, and Ogg offered a bottle of Jack Daniels to anyone who could give an explanation: that these prime numbers are precisely the prime factors of the size—a 54 digit number—of the monster sporadic finite group, an extraordinary and enormous non-reducible set of symmetries. The connection between elliptic curves and the monster group was totally unexpected and puzzling. Monstrous moonshine is the technical term for this coincidence and the many mathematical and physical developments that have followed it.
A subsequent observation was made by John McKay in the 1980’s that certain subsets in the monster group are in direct correspondence to nodes of the E8 Dynkin diagram, one of a family of graphs that are used in the classification of continuous symmetries. It was also observed that the next two largest sporadic groups admit similar relations to the E7 and E6 Dynkin diagrams. These observations offered insight into monstrous moonshine and effort was made to make these correspondences precise, an effort that is ongoing.
This project will contribute to that effort. Research will be made into an auxiliary group G that shares the correspondence between certain of its subsets and the nodes of the E7 Dynkin diagram, in a manner identical to that of the second largest sporadic group observed by McKay. This auxiliary group shares the characteristic features of the correspondence while having the advantage of being much smaller in size. The group G also plays a key role in umbral moonshine, which is a recently discovered counterpart to monstrous moonshine with connections to string theories compactified using K3 surfaces. The aim of the project will be to explain the E7 correspondence for G.