Umbral Moonshine
Abstract
We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by EguchiOoguriTachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vectorvalued mock modular forms studied recently by DabholkarMurthyZagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinitedimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of ConwayNorton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKayThompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.2779
 Bibcode:
 2012arXiv1204.2779C
 Keywords:

 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 11F22;
 11F37;
 11F46;
 11F50;
 20C34;
 20C35
 EPrint:
 124 pages, 75 tables