exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
exceptional Jordan superalgebra, $K_10$
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 $M_24$ with K3 surfaces, and which corresponds to the Niemeier lattice with the simplest root system $X = A_1^{24}$. As noted in 2010 by Eguchi, Ooguri, and Tachikawa, dimensions of some representations of $M_24$, the largest sporadic simple Mathieu group, are multiplicities of superconformal algebra characters in the K3 elliptic genus.
Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine (arXiv:1204.2779)
Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine and the Niemeier Lattices (arXiv:1307.5793)
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture (arXiv:1503.01472)
John F. R. Duncan, Sander Mack-Crane, Derived Equivalences of K3 Surfaces and Twined Elliptic Genera, (arXiv:1506.06198)
Miranda C. N. Cheng, Sarah Harrison, Umbral Moonshine and K3 Surfaces, (arXiv:1406.0619)
Shamit Kachru, Natalie Paquette, Roberto Volpato, 3D String Theory and Umbral Moonshine (arXiv:1603.07330)