Mathematics – Representation Theory
Scientific paper
2012-04-12
Mathematics
Representation Theory
118 pages, 76 tables
Scientific paper
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 which characterises the Jacobi forms arising; they are closely related to vector-valued mock modular forms belonging to a sequence 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.
Cheng Miranda C. N.
Duncan John F. R.
Harvey Jeffrey A.
No associations
LandOfFree
Umbral Moonshine does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Umbral Moonshine, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Umbral Moonshine will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-313101