nLab umbral moonshine

Idea

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.

Related entries

• Niemeier lattice

• Moonshine

• K3 surface

References

• 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)