Misc,
Umbral Moonshine
(2012)cite arxiv:1204.2779Comment: 124 pages, 75 tables; updated references and minor editing in v.2; minor editing and two new theorems in v.3.
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 Eguchi-Ooguri-Tachikawa 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
vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier 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 infinite-dimensional 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
Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions
appearing as McKay-Thompson 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.
