umbral moonshine



The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. It relates the 23 Niemeier lattices, even unimodular positive-definite lattices of rank 24 with non-trivial root systems, to mock theta functions.

Umbral moonshine is a generalization of the Mathieu moonshine phenomenon which relates representations of the Mathieu group M 24M_24 with K3 surfaces, and which corresponds to the Niemeier lattice with the simplest root system X=A 1 24X = A_1^{24}. As noted in 2010 by Eguchi, Ooguri, and Tachikawa, dimensions of some representations of M 24M_24, the largest sporadic simple Mathieu group, are multiplicities of superconformal algebra characters in the K3 elliptic genus.