New Moonshines
( A. Taormina)

Description

In 1979, Conway and Norton initiated the study of a mysterious relation (observed by John McKay) between the simplest modular function j(τ) and the largest sporadic simple finite group, and named this relation `Monstrous Moonshine'. That these two objects, pertaining to two a priori distinct areas of mathematics, could have anything in common came as a surprise, and it took thirteen years of research, culminating with the work of Borcherds, to unravel the depth of this relation and prove the conjecture. In 2010, the observation by Eguchi, Ooguri and Tachikawa (EOT) of a relation between a specific mock modular form and the sporadic simple finite group Mathieu 24 soon became known as `Mathieu Moonshine', due to its similarities with the original Monstrous Moonshine. But alongside these similarities, the new Moonshine offers interesting perspectives on a possible deep relation with String Theory. This observation, still largely unexplained in the string context, has triggered a flurry of publications that have refined and generalised the Mathieu Moonshine. Techniques borrowed from Monstrous Moonshine have provided precious clues and subsequently a proof that the sporadic group in the EOT conjecture is indeed Mathieu 24. Further work by Cheng, Harvey and Duncan shows that Mathieu Moonshine is one phenomenon in a family of correspondences between finite groups and vector-valued mock modular forms, leading to the concept of `Umbral Moonshine'.

In light of the above results, it is reasonable to believe that Monstrous Moonshine, Mathieu Moonshine and Umbral Moonshine could be understood as special cases of one structure including finite groups, automorphic forms of certain kinds and string theory. At present we have small hints that there should be some links between these new moonshines and string theory.

The project will explore the fascinating interplay between group theory and mock modular forms in the context of umbral moonshine.

Prerequisites

Resources

-A starting point is http://arxiv.org/pdf/1201.4140.pdf, The Largest Mathieu Group and (Mock) Automorphic Forms, by Cheng and Duncan.

-Read http://arxiv.org/pdf/1307.5793.pdf, Umbral Moonshine and Niemeier lattices by Cheng, Duncan and Harvey to learn about umbral moonshine.

-For a nice introduction to mock modular forms, you may wish to consult http://arxiv.org/pdf/1208.4074.pdf, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, by Dabholkar, Harvey and Murthy.